I asked this previously but the expert didn't actually answer my question, I'm specifically asking how u=u(t) works into the answer, also how to go from f(t,x) to f(s,x(s). Thank you! **Problem 5:**Use the Fundamental Theorem of Calculus to show that the function \( u = u(t) \) is a solution to the initial value problem\[x' = f(t, x)\]\[x(t_0) = x_0,\]if and only if \( u \) is a solution to the integral equation\[x(t) = x_0 + \int_{t_0}^{t} f(s, x(s)) \, ds.\]

Answered: Image /qna-images/answer/79632763-4a8f-4e48-b5c9-7e2560724356.jpg

5. Use the Fundamental Theorem of Calculus to show that the function u = u(t) is a solution to the initial value problem a' = f(t, x) x(to) = xo, if and only if u is a solution to the integral equation x(t) = xo + | f(s, 2(s)) ds.

5. Use the Fundamental Theorem of Calculus to show that the function u = u(t) is a solution to the initial value problem a' = f(t, x) x(to) = xo, if and only if u is a solution to the integral equation x(t) = xo + | f(s, 2(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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I asked this previously but the expert didn't actually answer my question, I'm specifically asking how u=u(t) works into the answer, also how to go from f(t,x) to f(s,x(s). Thank you!

**Problem 5:**

Use the Fundamental Theorem of Calculus to show that the function \( u = u(t) \) is a solution to the initial value problem

\[
x' = f(t, x)
\]
\[
x(t_0) = x_0,
\]

if and only if \( u \) is a solution to the integral equation

\[
x(t) = x_0 + \int_{t_0}^{t} f(s, x(s)) \, ds.
\]

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