ODE with Parameter: Is \phi(x,0) a Solution to y' = f(x,y,0)?

[dipole]

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?

 

[voko]

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

 

[dipole]

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]

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:
$\phi(x, 0)'= f(x, y, 0)$.