# ODE with parameter question

## Homework Statement

In a HW assignment, I'm given the ODE

$y' = f(x,y,\epsilon)$

and that $y = \phi(x,\epsilon)$is a solution to this equation.

I'm then asked, is $\phi(x,0)$ a solution to the equation

$y' = f(x,y,0)$

This result is used for the second part of the problem, and in the question I'm told I can just quote a well known theorem to explain why it's true, but I have no idea what theorem that might be. Any ideas, or maybe how to even prove it?

## Answers and Replies

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There is a theorem on the dependence of ODE solutions on parameters. I am sure it has been covered in your course of ODEs.

You would think so, but the professor constantly assigns HW that has little relevance to what we've actually done in lecture. Also we have no textbook to use as a reference.

HallsofIvy
Science Advisor
Homework Helper
Since the derivative is with respect to x, not $\epsilon$, we can write $\phi(x, \epsilon)'= f(x, y, \epsilon)$ and set $\epsilon= 0$ in that equation:
$\phi(x, 0)'= f(x, y, 0)$.