Exercise 2.4.1 Use Theorem 2.4.1 to show that each initial value problem u' = f(t,u), u(to) = has a unique solution for some t interval around to. Make sure to specify what f is, and why it has the required properties. uo (a) u' (t) = u(t) +3, u(0) = 3 (b) u' (t)=-u²(t) + sin(t), u(1) = 4 (c) u' (t) = 1/u(t), u(2) = 2 (d) u' (t)=ru(t) (1-u(t)/K), u(to) = uo (the logistic equation).
Exercise 2.4.1 Use Theorem 2.4.1 to show that each initial value problem u' = f(t,u), u(to) = has a unique solution for some t interval around to. Make sure to specify what f is, and why it has the required properties. uo (a) u' (t) = u(t) +3, u(0) = 3 (b) u' (t)=-u²(t) + sin(t), u(1) = 4 (c) u' (t) = 1/u(t), u(2) = 2 (d) u' (t)=ru(t) (1-u(t)/K), u(to) = uo (the logistic equation).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Do b and d
Transcribed Image Text:4 Exercises
Exercise 2.4.1 Use Theorem 2.4.1 to show that each initial value problem u' = f(t,u), u(to) =
uo has a unique solution for some t interval around to. Make sure to specify what f is, and why
it has the required properties.
(a) u'(t) = u(t)+3, u(0) = 3
(b) u'(t)=-u²(t) + sin(t), u(1) = 4
(c) u'(t)=1/u(t), u(2) = 2
(d) u' (t) = ru(t) (1-u(t)/K), u(to) = uo (the logistic equation).
oro very general form of Newton's law of cooling given by
Transcribed Image Text:rectangle in the tu plane (in more advanced texts weaker conditions on f are permitted). We
summarize these facts in the following theorem.
Theorem 2.4.1 - Existence and Uniqueness of Solutions to ODES. Let R denote a rectangle
a<t<b and c <u<d in the tu plane. Suppose that the function f(t, u) is continuous at each
point in R and that is continuous at each point in R. Then for any point t = to, u = uo in R, the
af
ди
ODE u' (t) = f(t,u(t)) has a unique solution with u(to) = uo on some interval to -8 <t<to + ₂
with 8₁> 0 and 8₂ > 0.
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