Show that x(t) is a solution of the IVP x' = f(x), x(0) = xo if and only if it satisfies the integral equation %3D x(t) = x0 + f(x(s)) ds.
Show that x(t) is a solution of the IVP x' = f(x), x(0) = xo if and only if it satisfies the integral equation %3D x(t) = x0 + f(x(s)) ds.
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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![Show that x(t) is a solution of the IVP x' = f(x), x(0) = x0 if and only
if it satisfies the integral equation
= Xo + f(x(s)) ds.
Consider the sequence of functions as defined in the proof of the Existence
and Uniqueness Theorem, i.e., Xo(t) = xo
Xn+1(t) = xo +
I f(xn(s)) ds, n = 0, 1, ....
Show that if Xn(t) is defined, continuous and belongs to the compact
set D for all t e [-a, a] and f e C'(D), then xn+1(t) is defined and
continuous for all t E -a, a].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F096e3465-5e66-41e8-8526-c018fcf73d9b%2F628dc6d2-9cab-48de-a8a4-30368ffd942f%2F0keec4m_processed.png&w=3840&q=75)
Transcribed Image Text:Show that x(t) is a solution of the IVP x' = f(x), x(0) = x0 if and only
if it satisfies the integral equation
= Xo + f(x(s)) ds.
Consider the sequence of functions as defined in the proof of the Existence
and Uniqueness Theorem, i.e., Xo(t) = xo
Xn+1(t) = xo +
I f(xn(s)) ds, n = 0, 1, ....
Show that if Xn(t) is defined, continuous and belongs to the compact
set D for all t e [-a, a] and f e C'(D), then xn+1(t) is defined and
continuous for all t E -a, a].
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