Differential Equations / Analysis

[moo5003]

Homework Statement

Determine all infinitely differentiable real functions f(x) that satisfy f''(x)=-f(x) and f(2x)=2f(x)f'(x) for all real x. How do you know that you have exhibited all of them?

The Attempt at a Solution

f''(x)=-f(x)

I want to say the solutions to such an equation would be of the form:
acos(x)+bsin(x), but its hard to prove that this form consists of all solutions.

In class the teacher said that there are some differential equation classes that would tell you that this equation has two indepdendent solutions and since my form has two independent variables (a,b) that it makes up all solutions, but I'm not allowed to use this fact as I have yet to take the class. So, my main concern before I introduce the second restriction of f(x)f'(x)2=f(2x), is how do I validate that my form spans all possible functions that satisfy f''(x)=-f(x).

I'm leaning toward using linear algebra in some capacity and having cos/sin as a basis though I'm not sure how to proceed with this line of thought.

 

[EnumaElish]

The first equation is simple harmonic motion with angular frequency of oscillation = 1. This equation has a general solution that is unique up to two constants, A and B. http://mathworld.wolfram.com/SimpleHarmonicMotion.html Use the second equation to solve for the constant(s). If you can determine both constants (A and B) then you will have found a special unique solution.