8.11. Lineare Differentialgleichungen 1. Ordnung


In diesem Abschnitt verallgemeinern wir das Stammfunktionenproblem. Die Aufgabe, zu einer vorgegebenen Funktion g eine Stammfunktion f zu finden, formulieren wir dazu neu: Löse die (Funktionen-)gleichung

f =g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaeyypa0Jaam4zaaaa@38D2@ .

Die Unbekannte ist jetzt eine Funktion, die mit ihrer 1. Ableitung in der Gleichung auftritt. Gleichungen dieser Art heißen Differentialgleichungen.

Wir beginnen unsere Untersuchungen mit Differentialgleichungen des folgenden Typs:

Definition:  Ist g: MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaacQdacqWIDesOcqGHsgIRcqWIDesOaaa@3C60@ eine Funktion und a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgIGiolabl2riHcaa@39C3@ , so nennen wir die Gleichung

f +af=g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaWGNbaaaa@3B85@
[8.11.1]

eine (normierte) lineare Differentialgleichung 1. Ordnung mit konstanten Koeffizienten (über MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@3759@ ).

Unter einer Lösung dieser Gleichung verstehen wir eine auf MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@3759@ differenzierbare Funktion f, also f D 1 () MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgIGiolaadseadaahaaWcbeqaaiaaigdaaaGccaGGOaGaeSyhHeQaaiykaaaa@3CDC@ , die die Gleichung [8.11.1] erfüllt.

Ist speziell die rechte Seite g=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg2da9iaaicdaaaa@3895@ , so nennt man die Gleichung [8.11.1] homogen.

Beachte: Unsere Gleichungen

  • sind normiert, weil f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaaaaa@36E0@ den Koeffizienten 1 besitzt. Das ist aber keine Einschränkung, denn jede allgemeine Gleichung m f +af=g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiqadAgagaqbaiabgUcaRiaadggacaWGMbGaeyypa0Jaam4zaaaa@3C77@ läßt sich im Fall m0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabgcMi5kaaicdaaaa@395C@ per Division durch m stets normieren. Falls m=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2da9iaaicdaaaa@389B@ , liegt gar keine Differentialgleichung vor.

  • sind linear, weil die Unbekannte f und ihre Ableitung f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaaaaa@36E0@ weder in einem Produkt noch in einer Potenz vorkommen.

  • sind von 1. Ordnung, weil f in ihnen nur bis zur ersten Ableitung auftritt.

  • haben konstante Koeffizienten, weil der (hier einzige) Koeffizient a konstant ist. Allgemeine Differentialgleichungen erlauben auch Funktionen als Koeffizienten.

    Für a=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iaaicdaaaa@388F@ beschreibt die Gleichung [8.11.1] offenbar das alte Stammfunktionenproblem.

  • betrachten wir als Differentialgleichungen über MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@3759@ . Alle Ergebnisse lassen sich i.w. aber auch für ein beliebiges Intervall formulieren und beweisen.

Beim Lösen von Differentialgleichungen sind drei Problemkreise angesprochen:

  1. Das Existenzproblem: Gibt es zu jeder rechten Seite g eine Lösung?

  2. Das Eindeutigkeitsproblem: Kann es zu einem g mehrere Lösungen geben?

  3. Das Regularitätsproblem: Wenn g mehrfach differenzierbar ist, sind dann auch die Lösungen mehrfach differenzierbar?

Bereits von den Stammfunktionen her wissen wir, dass 1. und 2. nicht positiv zu beantworten sind. Bei der Wahl der rechten Seite g wird man also eingeschränkt sein. Die Eindeutigkeit werden wir, wie bei den Stammfunktionen auch, durch eine Zusatzbedingung, die sog. Anfangsbedingung, erzwingen.

Zunächst lösen wir das Existenz- und das Eindeutigkeitsproblem für homogene Differentialgleichungen, und zwar i.w. bereits durch die folgende Bemerkung.

Bemerkung:  Für jede D 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaCaaaleqabaGaaGymaaaaaaa@379A@ -Funktion f gilt:

f +af=0f=f(0) e aX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaaIWaGaaGzbVlabgsDiBlaaywW7caWGMbGaeyypa0JaamOzaiaacIcacaaIWaGaaiykaiabgwSixlaadwgadaahaaWcbeqaaiabgkHiTiaadggacaWGybaaaaaa@4BCE@
[8.11.2]

Beweis:

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@37C6@ ":  Wir arbeiten mit einem kleinen Trick und berechnen zunächst die Ableitung der differenzierbaren Funktion f e aX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGMbaabaGaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaaaaaaa@3AAB@ gemäß Quotienten- und Kettenregel:

( f e aX ) = f e aX +af e aX e 2aX = f +af e aX =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaalaaabaGaamOzaaqaaiaadwgadaahaaWcbeqaaiabgkHiTiaadggacaWGybaaaaaakiqacMcagaqbaiabg2da9maalaaabaGabmOzayaafaGaeyyXICTaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaGccqGHRaWkcaWGHbGaamOzaiabgwSixlaadwgadaahaaWcbeqaaiabgkHiTiaadggacaWGybaaaaGcbaGaamyzamaaCaaaleqabaGaeyOeI0IaaGOmaiaadggacaWGybaaaaaakiabg2da9maalaaabaGabmOzayaafaGaey4kaSIaamyyaiaadAgaaeaacaWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaamiwaaaaaaGccqGH9aqpcaaIWaaaaa@5BEE@ .

Nach [7.9.7], einer Folgerung aus dem Mittelwertsatz, ist damit die auf dem Intervall MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@3759@ definierte Funktion f e aX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGMbaabaGaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaaaaaaa@3AAB@ konstant: f e aX =c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGMbaabaGaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaaaaOGaeyypa0Jaam4yaaaa@3CA3@ . Folgt:

f=c e aX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iaadogacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaamiwaaaaaaa@3ED3@ .

Offensichtlich ist dabei f(0)=c e a0 =c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaaIWaGaaiykaiabg2da9iaadogacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaeyyXICTaaGimaaaakiabg2da9iaadogaaaa@4505@ .

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@37C2@ ":  Ist f=f(0) e aX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iaadAgacaGGOaGaaGimaiaacMcacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaamiwaaaaaaa@40E9@ , so hat man:

f +af=f(0)(a e aX +a e aX )=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaWGMbGaaiikaiaaicdacaGGPaGaeyyXICTaaiikaiabgkHiTiaadggacaWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaamiwaaaakiabgUcaRiaadggacaWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaamiwaaaakiaacMcacqGH9aqpcaaIWaaaaa@4E37@ .
 

Beachte: Das Ergebnis läßt sich auch in der Sprache der linearen Algebra notieren.

  • Die Lösungsmenge der Gleichung f +af=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaaIWaaaaa@3B53@ besteht nach [8.11.2] aus allen Vielfachen von e aX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaaaaa@39B0@ , ist also gleich dem Erzeugnis von e aX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaaaaa@39B0@ :

    f +af=0f< e aX > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaaIWaGaaGzbVlabgsDiBlaaywW7caWGMbGaeyicI4SaeyipaWJaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaGccqGH+aGpaaa@491A@ .

    Sie ist damit ein Untervektorraum von D 1 () MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaCaaaleqabaGaaGymaaaakiaacIcacqWIDesOcaGGPaaaaa@3A70@ , und zwar der Dimension 1, da e aX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaaaaa@39B3@ linear unabhängig ist.

  • Führt man den linearen Differentialoperator  D X+a : D 1 ()F() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBaaaleaacaWGybGaey4kaSIaamyyaaqabaGccaGG6aGaamiramaaCaaaleqabaGaaGymaaaakiaacIcacqWIDesOcaGGPaGaeyOKH4QaamOraiaacIcacqWIDesOcaGGPaaaaa@4453@ durch die Festsetzung

    D X+a (f) f +af MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBaaaleaacaWGybGaey4kaSIaamyyaaqabaGccaGGOaGaamOzaiaacMcacqGH9aqpceWGMbGbauaacqGHRaWkcaWGHbGaamOzaaaa@4084@

    ein, so ist < e aX > MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyipaWJaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaGccqGH+aGpaaa@3BC9@ gerade der Kern von D X+a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBaaaleaacaWGybGaey4kaSIaamyyaaqabaaaaa@3983@ . Man beachte überdies, dass a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaamyyaaaa@37BC@ hier die Nullstelle des Polynoms X+a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiabgUcaRiaadggaaaa@388E@ ist.

[8.11.2] belegt, dass eine homogene Differentialgleichung unendliche viele Lösungen besitzt. Die angestrebte Eindeutigkeit erhalten wir nun durch die zusätzliche Forderung (Anfangsbedingung) f(0)=c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaaIWaGaaiykaiabg2da9iaadogaaaa@3AD5@ .

Bemerkung:  Für jedes c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabgIGiolabl2riHcaa@39C5@ hat die homogene Gleichung f +af=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaaIWaaaaa@3B53@ unter der Anfangsbedingung f(0)=c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaaIWaGaaiykaiabg2da9iaadogaaaa@3AD5@ genau eine Lösung:

f +af=0      f(0)=cf=c e aX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaaIWaGaaGjbVlabgEIizlaaysW7caWGMbGaaiikaiaaicdacaGGPaGaeyypa0Jaam4yaiaaywW7cqGHuhY2caaMf8UaamOzaiabg2da9iaadogacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaamiwaaaaaaa@5369@
[8.11.3]

Beweis:  Die Behauptung folgt direkt aus [8.11.2], und zwar über die Äquivalenz:

f=f(0) e aX       f(0)=cf=c e aX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iaadAgacaGGOaGaaGimaiaacMcacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaamiwaaaakiaaysW7cqGHNis2caaMe8UaamOzaiaacIcacaaIWaGaaiykaiabg2da9iaadogacaaMf8Uaeyi1HSTaaGzbVlaadAgacqGH9aqpcaWGJbGaeyyXICTaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaaaaa@5909@ .

Gelegentlich ist die Anfangsbedingung nicht für den Punkt 0 formuliert. [8.11.3] läßt sich jedoch in dieser Hinsicht leicht verallgemeinern, denn für beliebige b,c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiaacYcacaWGJbGaeyicI4SaeSyhHekaaa@3B5C@ gilt:

f +af=0      f(b)=cf=c e a(Xb) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaaIWaGaaGjbVlabgEIizlaaysW7caWGMbGaaiikaiaadkgacaGGPaGaeyypa0Jaam4yaiaaywW7cqGHuhY2caaMf8UaamOzaiabg2da9iaadogacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaaiikaiaadIfacqGHsislcaWGIbGaaiykaaaaaaa@56C6@
[8.11.4]

Beweis:  Wir setzen die Verschiebungen X+b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiabgUcaRiaadkgaaaa@388F@ und Xb MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwaiabgkHiTiaadkgaaaa@389A@ ein. Da gemäß Kettenregel die Gleichheit (f(X+b) ) = f (X+b) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgacqWIyiYBcaGGOaGaamiwaiabgUcaRiaadkgacaGGPaGabiykayaafaGaeyypa0JabmOzayaafaGaeSigI8MaaiikaiaadIfacqGHRaWkcaWGIbGaaiykaaaa@44A8@ gegeben ist, kann man mit [8.11.3] folgendermaßen argumentieren:

f +af=0      f(b)=c (f(X+b) ) +af(X+b)=0      f(X+b)(0)=c f(X+b)=c e aX f=c e a(Xb) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@86C3@
 

Beispiel:  

  • f +3f=0      f(0)=5f=5 e 3X MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaaG4maiaadAgacqGH9aqpcaaIWaGaaGjbVlabgEIizlaaysW7caWGMbGaaiikaiaaicdacaGGPaGaeyypa0JaaGynaiaaywW7cqGHuhY2caaMf8UaamOzaiabg2da9iaaiwdacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaaIZaGaamiwaaaaaaa@52C5@

  • f 7f=0      f(2)=4f=4 e 7(X2) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaeyOeI0IaaG4naiaadAgacqGH9aqpcaaIWaGaaGjbVlabgEIizlaaysW7caWGMbGaaiikaiaaikdacaGGPaGaeyypa0JaeyOeI0IaaGinaiaaywW7cqGHuhY2caaMf8UaamOzaiabg2da9iabgkHiTiaaisdacqGHflY1caWGLbWaaWbaaSqabeaacaaI3aGaaiikaiaadIfacqGHsislcaaIYaGaaiykaaaaaaa@56C7@

  • f +f=0      f(8)=2f=2 e (X+8) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamOzaiabg2da9iaaicdacaaMe8Uaey4jIKTaaGjbVlaadAgacaGGOaGaeyOeI0IaaGioaiaacMcacqGH9aqpcaaIYaGaaGzbVlabgsDiBlaaywW7caWGMbGaeyypa0JaaGOmaiabgwSixlaadwgadaahaaWcbeqaaiabgkHiTiaacIcacaWGybGaey4kaSIaaGioaiaacMcaaaaaaa@5537@

Anwendungsbeispiele belegen die Bedeutung des Differentialgleichungen im naturwissenschaftlichen Bereich. Einige Beispiele zu Wachstumsprozessen geben hier einen ersten Eindruck.

Interessanterweise kann man das Lösungsverfahren für homogene Differentialgleichungen auch auf Gleichungen mit nicht-konstanten Koeffizienten übertragen. Der Beweis orientiert sich dabei an [8.11.2]. Man beachte in diesem Zusammenhang, dass aX MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadIfaaaa@37AC@ eine Stammfunktion zur konstanten Funktion a ist.

Bemerkung:  Ist s eine Stammfunktion zur integrierbaren Funktion rI() MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabgIGiolaadMeacaGGOaGaeSyhHeQaaiykaaaa@3BFB@ und b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabgIGiolabl2riHcaa@39C4@ , so gilt für cb e s(0) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabg2da9iaadkgacqGHflY1caWGLbWaaWbaaSqabeaacaWGZbGaaiikaiaaicdacaGGPaaaaaaa@3F2A@ :

f +rf=0      f(0)=bf=c e s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamOCaiaadAgacqGH9aqpcaaIWaGaaGjbVlabgEIizlaaysW7caWGMbGaaiikaiaaicdacaGGPaGaeyypa0JaamOyaiaaywW7cqGHuhY2caaMf8UaamOzaiabg2da9iaadogacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGZbaaaaaa@52AE@
[8.11.5]

Beweis:  ?

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4naaa@37C6@ ":  Da ( f e s ) = f e s +rf e s e 2s = f +rf e s =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikamaalaaabaGaamOzaaqaaiaadwgadaahaaWcbeqaaiabgkHiTiaadohaaaaaaOGabiykayaafaGaeyypa0ZaaSaaaeaaceWGMbGbauaacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGZbaaaOGaey4kaSIaamOCaiaadAgacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGZbaaaaGcbaGaamyzamaaCaaaleqabaGaeyOeI0IaaGOmaiaadohaaaaaaOGaeyypa0ZaaSaaaeaaceWGMbGbauaacqGHRaWkcaWGYbGaamOzaaqaaiaadwgadaahaaWcbeqaaiabgkHiTiaadohaaaaaaOGaeyypa0JaaGimaaaa@5819@ , ist die auf MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@3759@ differenzierbare Funktion f e s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGMbaabaGaamyzamaaCaaaleqabaGaeyOeI0Iaam4Caaaaaaaaaa@39E0@ konstant. Es gibt also ein k, so dass

f=k e s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iaadUgacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGZbaaaaaa@3E10@ .

Aus b=f(0)=k e s(0) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabg2da9iaadAgacaGGOaGaaGimaiaacMcacqGH9aqpcaWGRbGaeyyXICTaamyzamaaCaaaleqabaGaeyOeI0Iaam4CaiaacIcacaaIWaGaaiykaaaaaaa@4423@ folgt schließlich: k=b e s(0) =c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2da9iaadkgacqGHflY1caWGLbWaaWbaaSqabeaacaWGZbGaaiikaiaaicdacaGGPaaaaOGaeyypa0Jaam4yaaaa@412A@ .

" MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=qqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi0HWnaaa@37C2@ ":  Ist f=c e s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iaadogacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGZbaaaaaa@3E08@ , so hat man:

f +rf=rc e s +rc e s =0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamOCaiaadAgacqGH9aqpcqGHsislcaWGYbGaam4yaiabgwSixlaadwgadaahaaWcbeqaaiabgkHiTiaadohaaaGccqGHRaWkcaWGYbGaam4yaiabgwSixlaadwgadaahaaWcbeqaaiabgkHiTiaadohaaaGccqGH9aqpcaaIWaaaaa@4C97@

und: f(0)=c e s(0) =b e s(0) e s(0) =b MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaaIWaGaaiykaiabg2da9iaadogacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGZbGaaiikaiaaicdacaGGPaaaaOGaeyypa0JaamOyaiabgwSixlaadwgadaahaaWcbeqaaiaadohacaGGOaGaaGimaiaacMcaaaGccqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGZbGaaiikaiaaicdacaGGPaaaaOGaeyypa0JaamOyaaaa@53EB@ .

Da cos eine Stammfunktion zu sin MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaci4CaiaacMgacaGGUbaaaa@39AE@ und 3 e cos(0) =3e MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgwSixlaadwgadaahaaWcbeqaaiGacogacaGGVbGaai4CaiaacIcacaaIWaGaaiykaaaakiabg2da9iaaiodacaWGLbaaaa@41A4@ ist, haben wir mit

  • f sinf=0      f(0)=3f=3e e cos =3 e 1cos MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaeyOeI0Iaci4CaiaacMgacaGGUbGaeyyXICTaamOzaiabg2da9iaaicdacaaMe8Uaey4jIKTaaGjbVlaadAgacaGGOaGaaGimaiaacMcacqGH9aqpcaaIZaGaaGzbVlabgsDiBlaaywW7caWGMbGaeyypa0JaaG4maiaadwgacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislciGGJbGaai4BaiaacohaaaGccqGH9aqpcaaIZaGaeyyXICTaamyzamaaCaaaleqabaGaaGymaiabgkHiTiGacogacaGGVbGaai4Caaaaaaa@62FD@

ein Beispiel zu [8.11.5] notiert.

Wir wenden uns nun den inhomogenen Gleichungen zu. Wie zu Beginn bereits erwähnt, ist hier Wahl der rechten Seite g nicht beliebig. Zufriedenstellende Ergebniss erhalten wir allerdings, wenn die rechte Seite stetig ist. Ein Beispiel im Abschnitt über Stammfunktionen zeigt jedoch, dass auch ein unstetiges g zu einer lösbaren Differentialgleichung führen kann.

Die folgende Bemerkung ist eine direkte Parallele zu [8.11.2]. Auch sie löst i.w. bereits das Existenz- und das Eindeutigkeitsproblem. Wesentliches Hilfsmittel ist dabei das Faltungsprodukt [8.10.1].

Bemerkung:  Für jede stetige Funktion g: MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaacQdacqWIDesOcqGHsgIRcqWIDesOaaa@3C60@ gilt:

f +af=gf=f(0) e aX + e aX g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaWGNbGaaGzbVlabgsDiBlaaywW7caWGMbGaeyypa0JaamOzaiaacIcacaaIWaGaaiykaiabgwSixlaadwgadaahaaWcbeqaaiabgkHiTiaadggacaWGybaaaOGaey4kaSIaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaGccqGHxiIkcaWGNbaaaa@5295@
[8.11.6]

Beweis:  Nach [8.10.9] ist ( e aX g ) =ga e aX g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadwgadaahaaWcbeqaaiabgkHiTiaadggacaWGybaaaOGaey4fIOIaam4zaiqacMcagaqbaiabg2da9iaadEgacqGHsislcaWGHbGaeyyXICTaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaGccqGHxiIkcaWGNbaaaa@48B5@ , also:

g=( e aX g ) +a e aX g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg2da9iaacIcacaWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaamiwaaaakiabgEHiQiaadEgaceGGPaGbauaacqGHRaWkcaWGHbGaeyyXICTaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaGccqGHxiIkcaWGNbaaaa@48AA@ .[1]

Damit können wir unsere Behauptung auf [8.11.2] zurück führen (beachte dabei auch [8.10.2]):

f +af=g f +afg=0 (f e aX g ) +a(f e aX g)=0 f e aX g=(f e aX g)(0) e aX f=f(0) e aX + e aX g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8CCB@

Beachte:

  • Nach [1] ist e aX g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaGccqGHxiIkcaWGNbaaaa@3B95@ eine spezielle Lösung der inhomogenen Gleichung f +af=g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaWGNbaaaa@3B85@ , denn sie erfüllt die Anfangsbedingung f(0)=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaaIWaGaaiykaiabg2da9iaaicdaaaa@3AA7@ . [8.11.6] zeigt nun, dass man die gesamte Lösungsmenge von f +af=g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaWGNbaaaa@3B85@ erhält, indem man die spezielle Lösung e aX g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaGccqGHxiIkcaWGNbaaaa@3B95@ zu jeder Lösung der homogenen Gleichung f +af=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaaIWaaaaa@3B53@ addiert, also den 1-dimensionalen affinen Unterraum

    e aX g+< e aX >= e aX g+Ker D X+a MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaGccqGHxiIkcaWGNbGaaGPaVlabgUcaRiabgYda8iaadwgadaahaaWcbeqaaiabgkHiTiaadggacaWGybaaaOGaeyOpa4Jaeyypa0JaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaGccqGHxiIkcaWGNbGaaGPaVlabgUcaRiaadUeacaWGLbGaamOCaiaayIW7caWGebWaaSbaaSqaaiaadIfacqGHRaWkcaWGHbaabeaaaaa@54DD@

    von D 1 () MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaCaaaleqabaGaaGymaaaakiaacIcacqWIDesOcaGGPaaaaa@3A70@ bildet. Dieser Begriff und seine Notation stammen aus der linearen Algebra.

Wie bereits im homogenen Fall erzwingen wir die Eindeutigkeit der Lösung auch hier über eine Anfangsbedingung.

Bemerkung:  Für jedes c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabgIGiolabl2riHcaa@39C8@ und jede stetig Funktion g hat die inhomogene Gleichung f +af=g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaWGNbaaaa@3B88@ unter der Anfangsbedingung f(0)=c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaaIWaGaaiykaiabg2da9iaadogaaaa@3AD8@ genau eine Lösung:

f +af=g      f(0)=cf=c e aX + e aX g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaWGNbGaaGjbVlabgEIizlaaysW7caWGMbGaaiikaiaaicdacaGGPaGaeyypa0Jaam4yaiaaywW7cqGHuhY2caaMf8UaamOzaiabg2da9iaadogacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaamiwaaaakiabgUcaRiaadwgadaahaaWcbeqaaiabgkHiTiaadggacaWGybaaaOGaey4fIOIaam4zaaaa@5A36@
[8.11.7]

Beweis:  Ähnlich wie in [8.11.3] genügt hier ebenfalls der Hinweis auf die Äquivalenz

f=f(0) e aX + e aX g      f(0)=cf=c e aX + e aX g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iaadAgacaGGOaGaaGimaiaacMcacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaamiwaaaakiabgUcaRiaadwgadaahaaWcbeqaaiabgkHiTiaadggacaWGybaaaOGaey4fIOIaam4zaiaaysW7cqGHNis2caaMe8UaamOzaiaacIcacaaIWaGaaiykaiabg2da9iaadogacaaMf8Uaeyi1HSTaaGzbVlaadAgacqGH9aqpcaWGJbGaeyyXICTaamyzamaaCaaaleqabaGaeyOeI0IaamyyaiaadIfaaaGccqGHRaWkcaWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaamiwaaaakiabgEHiQiaadEgaaaa@6632@ .

Auch bei inhomogenen Gleichungen kann man die Anfangsbedingung f(0)=c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaaIWaGaaiykaiabg2da9iaadogaaaa@3AD8@ durch die allgemeinere f(b)=c MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWGIbGaaiykaiabg2da9iaadogaaaa@3B05@ austauschen. Das Ergebnis läßt sich allerdings nicht mehr so kompakt notieren wie im homogenen Fall:

f +af=g      f(b)=cf=c e a(Xb) +( e aX (g(X+b)))(Xb) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaWGNbGaaGjbVlabgEIizlaaysW7caWGMbGaaiikaiaadkgacaGGPaGaeyypa0Jaam4yaiaaywW7cqGHuhY2caaMf8UaamOzaiabg2da9iaadogacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaaiikaiaadIfacqGHsislcaWGIbGaaiykaaaakiabgUcaRiaacIcacaWGLbWaaWbaaSqabeaacqGHsislcaWGHbGaamiwaaaakiabgEHiQiaacIcacaWGNbGaeSigI8MaaiikaiaadIfacqGHRaWkcaWGIbGaaiykaiaacMcacaGGPaGaeSigI8MaaiikaiaadIfacqGHsislcaWGIbGaaiykaaaa@6ABF@
[8.11.8]

Beweis:  Wir gehen wie im Beweis zu [8.11.4] vor und wenden dabei das Ergebnis [8.11.7] auf die stetige Funktion g(X+b) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiablIHiVjaacIcacaWGybGaey4kaSIaamOyaiaacMcaaaa@3C11@ an:

f +af=g      f(b)=c (f(X+b) ) +af(X+b)=g(X+b)      f(X+b)(0)=c f(X+b)=c e aX + e aX (g(X+b)) f=c e a(Xb) +( e aX (g(X+b)))(Xb) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@AD4A@
 

Beispiele zu inhomogenen Differentialgleichungen sind natürlich ungleich aufwändiger, denn hier müssen Faltungsprodukte ermittelt werden.

Beispiel:  

  •  

    f +5f= e 2X       f(0)=0 f= e 5X e 2X = 1 7 ( e 2X e 5X ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaaqaaaqaaiqadAgagaqbaiabgUcaRiaaiwdacaWGMbGaeyypa0JaamyzamaaCaaaleqabaGaaGOmaiaadIfaaaGccaaMe8Uaey4jIKTaaGjbVlaadAgacaGGOaGaaGimaiaacMcacqGH9aqpcaaIWaaabaGaeyi1HSnabaGaamOzaiabg2da9iaadwgadaahaaWcbeqaaiabgkHiTiaaiwdacaWGybaaaOGaey4fIOIaamyzamaaCaaaleqabaGaaGOmaiaadIfaaaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaI3aaaaiaacIcacaWGLbWaaWbaaSqabeaacaaIYaGaamiwaaaakiabgkHiTiaadwgadaahaaWcbeqaaiabgkHiTiaaiwdacaWGybaaaOGaaiykaaaaaaa@5D9D@

    Faltungsprodukt berechnen 

     i

    e 5X e 2X (x) = 0 x e 5(xX) e 2X = e 5x 0 x e 7X = e 5x 1 7 e 7X | 0 x = 1 7 e 5x ( e 7x 1) = 1 7 ( e 2x e 5x ) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8265@

  •  

    f +f=X      f(0)=3 f=3 e X + e X X=3 e X +X1+ e X =2 e X +X+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@7033@

    Faltungsprodukt berechnen 

     i

    e X X(x) = 0 x e (xX) X = e x 0 x e X X = e x ( e X X | 0 x 0 x e X ) partielle Integration = e x ( e x x e x +1) =x1+ e x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@970E@

  •  

    f +2f=sin      f(3)=1 f= e 2(X3) +( e 2X (sin(X+3)))(X3) = e 2(X3) +( 2 5 (sin(X+3)sin(3) e 2X ) 1 5 (cos(X+3)cos(3) e 2X ))(X3) = e 2(X3) + 2 5 (sinsin(3) e 2(X3) ) 1 5 (coscos(3) e 2(X3) ) = 2 5 sin 1 5 cos+ 52sin(3)+cos(3) 5 e 2(X3) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@FAAA@

    Faltungsprodukt berechnen 

     i

    Wir integrieren zweimal partiell und erhalten zunächst:

    e 2X sin(X+3)(x) = 0 x e 2(xX) sin(X+3) = 1 2 e 2(xX) sin(X+3) | 0 3 1 2 0 x e 2(xX) cos(X+3) = 1 2 e 2(xX) sin(X+3) | 0 3 1 2 ( 1 2 e 2(xX) cos(X+3) | 0 x + 1 2 0 x e 2(xX) sin(X+3) ) = 1 2 e 2(xX) sin(X+3) | 0 3 1 4 e 2(xX) cos(X+3) | 0 x 1 4 0 x e 2(xX) sin(X+3) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabuGaaaaabaaabaGaamyzamaaCaaaleqabaGaeyOeI0IaaGOmaiaadIfaaaGccqGHxiIkciGGZbGaaiyAaiaac6gacqWIyiYBcaGGOaGaamiwaiabgUcaRiaaiodacaGGPaGaaiikaiaadIhacaGGPaaabaGaeyypa0dabaWaa8qCaeaacaWGLbWaaWbaaSqabeaacqGHsislcaaIYaGaaiikaiaadIhacqGHsislcaWGybGaaiykaaaakiabgwSixlGacohacaGGPbGaaiOBaiaacIcacaWGybGaey4kaSIaaG4maiaacMcaaSqaaiaaicdaaeaacaWG4baaniabgUIiYdaakeaacqGH9aqpaeaadaWcaaqaaiaaigdaaeaacaaIYaaaaiabgwSixlaadwgadaahaaWcbeqaaiabgkHiTiaaikdacaGGOaGaamiEaiabgkHiTiaadIfacaGGPaaaaOGaeyyXICTaci4CaiaacMgacaGGUbGaaiikaiaadIfacqGHRaWkcaaIZaGaaiykaiaacYhadaqhaaWcbaGaaGimaaqaaiaaiodaaaGccqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaamaapehabaGaamyzamaaCaaaleqabaGaeyOeI0IaaGOmaiaacIcacaWG4bGaeyOeI0IaamiwaiaacMcaaaGccqGHflY1ciGGJbGaai4BaiaacohacaGGOaGaamiwaiabgUcaRiaaiodacaGGPaaaleaacaaIWaaabaGaamiEaaqdcqGHRiI8aaGcbaGaeyypa0dabaWaaSaaaeaacaaIXaaabaGaaGOmaaaacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaaIYaGaaiikaiaadIhacqGHsislcaWGybGaaiykaaaakiabgwSixlGacohacaGGPbGaaiOBaiaacIcacaWGybGaey4kaSIaaG4maiaacMcacaGG8bWaa0baaSqaaiaaicdaaeaacaaIZaaaaOGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGOaWaaSaaaeaacaaIXaaabaGaaGOmaaaacqGHflY1caWGLbWaaWbaaSqabeaacqGHsislcaaIYaGaaiikaiaadIhacqGHsislcaWGybGaaiykaaaakiabgwSixlGacogacaGGVbGaai4CaiaacIcacaWGybGaey4kaSIaaG4maiaacMcacaGG8bWaa0baaSqaaiaaicdaaeaacaWG4baaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaadaWdXbqaaiaadwgadaahaaWcbeqaaiabgkHiTiaaikdacaGGOaGaamiEaiabgkHiTiaadIfacaGGPaaaaOGaeyyXICTaci4CaiaacMgacaGGUbGaaiikaiaadIfacqGHRaWkcaaIZaGaaiykaaWcbaGaaGimaaqaaiaadIhaa0Gaey4kIipakiaacMcaaeaacqGH9aqpaeaadaWcaaqaaiaaigdaaeaacaaIYaaaaiabgwSixlaadwgadaahaaWcbeqaaiabgkHiTiaaikdacaGGOaGaamiEaiabgkHiTiaadIfacaGGPaaaaOGaeyyXICTaci4CaiaacMgacaGGUbGaaiikaiaadIfacqGHRaWkcaaIZaGaaiykaiaacYhadaqhaaWcbaGaaGimaaqaaiaaiodaaaGccqGHsisldaWcaaqaaiaaigdaaeaacaaI0aaaaiabgwSixlaadwgadaahaaWcbeqaaiabgkHiTiaaikdacaGGOaGaamiEaiabgkHiTiaadIfacaGGPaaaaOGaeyyXICTaci4yaiaac+gacaGGZbGaaiikaiaadIfacqGHRaWkcaaIZaGaaiykaiaacYhadaqhaaWcbaGaaGimaaqaaiaadIhaaaGccqGHsisldaWcaaqaaiaaigdaaeaacaaI0aaaamaapehabaGaamyzamaaCaaaleqabaGaeyOeI0IaaGOmaiaacIcacaWG4bGaeyOeI0IaamiwaiaacMcaaaGccqGHflY1ciGGZbGaaiyAaiaac6gacaGGOaGaamiwaiabgUcaRiaaiodacaGGPaaaleaacaaIWaaabaGaamiEaaqdcqGHRiI8aaaaaaa@156E@

    Damit ergibt sich:

    0 x e 2(xX) sin(X+3) = 4 5 ( 1 2 e 2(xX) sin(X+3) | 0 x 1 4 e 2(xX) cos(X+3) | 0 x ) = 2 5 (sin(x+3) e 2x sin(3)) 1 5 (cos(x+3) e 2x cos(3)) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@A926@

Das zu Beginn erwähnte Regularitätsproblem beantworten wir jetzt positiv: Jede Lösung f ist um eine Differenzierbarkeitsklasse besser als die rechte Seite g.

Bemerkung:  Ist f eine Lösung der Gleichung f +af=g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaWGNbaaaa@3B85@ , so gilt für jedes n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLoaaCaaaleqabaGaey4fIOcaaaaa@3AE8@ :

  1. g D n ()f D n+1 () MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabgIGiolaadseadaahaaWcbeqaaiaad6gaaaGccaGGOaGaeSyhHeQaaiykaiaaywW7cqGHshI3caaMf8UaamOzaiabgIGiolaadseadaahaaWcbeqaaiaad6gacqGHRaWkcaaIXaaaaOGaaiikaiabl2riHkaacMcaaaa@4B56@
[8.11.9]
  1. g C n ()f C n+1 () MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabgIGiolaadoeadaahaaWcbeqaaiaad6gaaaGccaGGOaGaeSyhHeQaaiykaiaaywW7cqGHshI3caaMf8UaamOzaiabgIGiolaadoeadaahaaWcbeqaaiaad6gacqGHRaWkcaaIXaaaaOGaaiikaiabl2riHkaacMcaaaa@4B54@
[8.11.10]
  1. g C ()f C () MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabgIGiolaadoeadaahaaWcbeqaaiabg6HiLcaakiaacIcacqWIDesOcaGGPaGaaGzbVlabgkDiElaaywW7caWGMbGaeyicI4Saam4qamaaCaaaleqabaGaeyOhIukaaOGaaiikaiabl2riHkaacMcaaaa@4AB3@
[8.11.11]

Beweis:  3. ist eine direkte Folgerung aus 1. Die Aussagen 1. und 2. beweisen wir simultan per Induktion und beachten dabei die Gleichung

f =gaf MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaeyypa0Jaam4zaiabgkHiTiaadggacaWGMbaaaa@3B90@ [2]
  • " n=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2da9iaaigdaaaa@389D@ ":  Als Lösung der Differentialgleichung f +af=g MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaey4kaSIaamyyaiaadAgacqGH9aqpcaWGNbaaaa@3B85@ ist f differenzierbar. Ist nun g eine D 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaCaaaleqabaGaaGymaaaaaaa@379A@ -Funktion, so trifft dies nach [2] auch auf f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaaaaa@36E0@ zu, wobei

    f = g a f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaagaGaeyypa0Jabm4zayaafaGaeyOeI0IaamyyaiqadAgagaqbaaaa@3BA9@ .

    Insbesondere ist f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaaaaa@36E0@ stetig, so dass auch f MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaagaaaaa@36E1@ stetig ist, falls g eine C 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCaaaleqabaGaaGymaaaaaaa@3799@ -Funktion ist. Also ist f eine D 2 / C 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaCaaaleqabaGaaGOmaaaakiaac+cacaWGdbWaaWbaaSqabeaacaaIYaaaaaaa@3A09@ -Funktion.

  • " n      n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaaysW7cqGHshI3caaMe8UaamOBaiabgUcaRiaaigdaaaa@3EE3@ ":  Sei jetzt g eine D n+1 / C n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaCaaaleqabaGaamOBaiabgUcaRiaaigdaaaGccaGGVaGaam4qamaaCaaaleqabaGaamOBaiabgUcaRiaaigdaaaaaaa@3DB1@ -Funktion, also erst recht eine D n / C n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaCaaaleqabaGaamOBaaaakiaac+cacaWGdbWaaWbaaSqabeaacaWGUbaaaaaa@3A77@ -Funktion. Gemäß Induktionsvoraussetzung ist f dann ( n+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgUcaRiaaigdaaaa@3879@ )-mal (stetig) differenzierbar, wobei nach [2]

    f (n+1) = g (n) a f (n) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaaCaaaleqabaGaaiikaiaad6gacqGHRaWkcaaIXaGaaiykaaaakiabg2da9iaadEgadaahaaWcbeqaaiaacIcacaWGUbGaaiykaaaakiabgkHiTiaadggacaWGMbWaaWbaaSqabeaacaGGOaGaamOBaiaacMcaaaaaaa@44A0@ .

    Damit aber ist f D n+2 / C n+2 () MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabgIGiolaadseadaahaaWcbeqaaiaad6gacqGHRaWkcaaIYaaaaOGaai4laiaadoeadaahaaWcbeqaaiaad6gacqGHRaWkcaaIYaaaaOGaaiikaiabl2riHkaacMcaaaa@42F5@ gesichert, denn g (n) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaCaaaleqabaGaaiikaiaad6gacaGGPaaaaaaa@394E@ ist eine D 1 / C 1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrVepeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaCaaaleqabaGaaGymaaaakiaac+cacaWGdbWaaWbaaSqabeaacaaIXaaaaaaa@3A07@ -Funktion.


8.10. 8.12.