By Earl A. Coddington

ISBN-10: 0486659429

ISBN-13: 9780486659428

"Written in an admirably cleancut and low-priced style." — Mathematical Review. a radical, systematic first path in trouble-free differential equations for undergraduates in arithmetic and technological know-how, requiring in simple terms easy calculus for a historical past, and together with many workouts designed to advance students' strategy in fixing equations. With difficulties and solutions. Index.

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Extra resources for An Introduction to Ordinary Differential Equations (Dover Books on Mathematics)

Example text

Die zentrale Frage, wann diese Foige gegen eine Losung der Gleichung (4) konvergiert, wird beantwortet durch den folgenden IX. Fixpunktsatz. Es sei D eine nichtleere, abgeschlossene Tei/menge eines Banach-Raumes B. Der Operator T sei auf D erkliirt und bi/de D in sich ab, T(D) C D. Er genuge in D einer Lipschitzbedingung mit einer Konstante k < 1: IITx-Tyll~kllx-YII fUr x,YED. (3) Dann hat die Gleichung (4) in D genau eine LOsung x = x. Bildet man, ausgehend von einem beliebigen Element Xo E D, die "sukzessiven Approximationen" x.

1st y eine negative Losung der Bernoulli-Differentialgleiehung, so ist u = - y eine Losung der Bernoulli-Differentialgleichung mit h ersetzt durch - h. Entspreehend ist v = u 1 - . eine positive Losung von (9) mit h ersetzt durch -h, also z = - v eine negative Losung der ursprtingliehen Gleichung (9). ) der Bernoulli-Differentialgleichung. Aus einer Losung z von (9) beliebigen Vorzeiehens ergibt sieh eine Losung I y(x) = Iz(x)lT=7 . sgnz der Bernoulli-Differentialgleiehung, und auf diese Weise erhalt man (abgesehen eventuell von y == 0) aile Losungen.

Dabei ist es sinnvoll + h2 >0 (3) vorauszusetzen (ist nlimlich 9 == h == 0 in einem Gebiet D, so ist jede in D verlaufende Kurve x = x(t), y = y(t) Losung von (2»). Ferner wird von einer Losung verlangt, daB x(t), y(t) stetig differenzierbar und (4) ist, d. h. daB ein "glattes Kurvenstiick" vorliegt. Auch diese Voraussetzung ist natiirlich. Sie schlieBt z. B. Losungen der Form x(t) = const, y(t) = const aus. Ferner wird durch sie garantiert, daB die Kurve lokal (d. h. in einer Umgebung eines jeden Kurvenpunktes) explizit in der Form y = 4J(x) oder x = 1p(y) mit 4J bzw.

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An Introduction to Ordinary Differential Equations (Dover Books on Mathematics) by Earl A. Coddington

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