1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Analysis - solutions to differential equations

  1. Apr 18, 2010 #1
    1. The problem statement, all variables and given/known data

    Suppose that f:R->R is twice differentiable and that f''(x) + f(x) = 0, f(0)=0 and f'(0)=0
    Prove that f'(x) = f(x) = 0 for all x

    2. Relevant equations

    3. The attempt at a solution

    I can solve this using methods from calulus, using an auxillary equation and the boundary conditions. However, I am unsure how to go about it as a piece of pure maths. In examples in my notes I have needed to use the 'Identity Theorem', a corollary of the Mean Value Theorem, stating if f: (a,b) -> R is differentiable and satisfies f'(t) = 0 for all t in (a,b) then f is constant on (a,b). However, I am unsure whether this is the correct method in this case, and if it is, how to use it.

    Thanks :)
  2. jcsd
  3. Apr 18, 2010 #2
    Consider the derivative of the function [itex]g = (f')^2 + f^2,[/itex] apply the "Identity Theorem", and use the initial conditions.
  4. Apr 19, 2010 #3
    Thanks! That makes a lot of sense.

    The next part of the question is a more general version: If g is twice differentiable and satisfies h''(x) + h(x) = 0 prove that h(x) = Acosx +Bsinx
    Using your advice, I can show h'(x)2+h(x)2 = constant
    I see that this looks a bit like Pythagoras but am not sure how I would prove that it involves sin and cos.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook