• Support PF! Buy your school textbooks, materials and every day products Here!

ODE with parameter question

  • Thread starter dipole
  • Start date
  • #1
537
146

Homework Statement



In a HW assignment, I'm given the ODE

[itex] y' = f(x,y,\epsilon) [/itex]

and that [itex] y = \phi(x,\epsilon) [/itex]is a solution to this equation.

I'm then asked, is [itex]\phi(x,0)[/itex] a solution to the equation

[itex] y' = f(x,y,0) [/itex]

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

  • #2
6,054
390
There is a theorem on the dependence of ODE solutions on parameters. I am sure it has been covered in your course of ODEs.
 
  • #3
537
146
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.
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,770
911
Since the derivative is with respect to x, not [itex]\epsilon[/itex], we can write [itex]\phi(x, \epsilon)'= f(x, y, \epsilon)[/itex] and set [itex]\epsilon= 0[/itex] in that equation:
[itex]\phi(x, 0)'= f(x, y, 0)[/itex].
 

Related Threads for: ODE with parameter question

Replies
3
Views
988
  • Last Post
Replies
6
Views
2K
Replies
1
Views
3K
Replies
0
Views
978
Replies
3
Views
2K
  • Last Post
Replies
1
Views
922
Top