Determine function with given properties

[phixmin]

Homework Statement

Determine all two time differentiable functions $f: \mathbb R \rightarrow \mathbb R$ that have the following properties:

$f'' = f, f(0) = 0,$ and $f'(0)=1$

The Attempt at a Solution

I've tried sinx, cosx, e^x, ln and linear combinations of them but I haven't found even one. Since the question says determine "all" I assume that there is a group of functions that hold these properties. Can anyone help me out here?

 

[Orodruin]

I've tried sinx, cosx, e^x

As Yoda would say: "Try not. Do, or do not. There is no try."

What you have is a differential equation $f'' = f$. First solve the differential equation by finding the general solution and then determine any constants by adapting the solution to the boundary conditions.

Also, "all" does not generally mean that there are more than a finite number - or even zero - but if it is zero you should have an argument for why it would be so. Typically, the general solution to a second order differential equation will have two integration constants that need to be determined through the boundary conditions.

 

[phixmin]

Thanks for the message! Unfortunately integration and differential equations are not yet introduced in the math course. We are only up to differentiating (it is a bachelor's math course for math students). Do you have any other ideas?

 

[Orodruin]

I do not see how you are supposed to solve it generally if you are not asked to solve the differential equation. However, you can solve it by Taylor expansion. (Since the function is twice differentiable and $f’’ = f$, it is infinitely differentiable.) Write down the Taylor expansion, insert it into the differential equation, and start identifying terms. See if you can figure out what function it is the Taylor expansion of.

 

[phixmin]

Ah very nice tip. I tried to do so and just got:

$\sum\limits_{n=0}^{\infty} a_n(x-a)^n = \sum\limits_{j=0}^{\infty} a_j*j*(j-1)(x-a)^{j-2}$

If I set a_n equal to the constant out in front of the left side, and the polynomials setting n = j-2...it doesn't bring me much further. I'm missing something.

 

[phixmin]

I also set both sides as "n" instead of using j and got:

$\sum\limits_{n=0}^{\infty} a_n(x-a)^n - a_n*n*(n-1)(x-a)^{n-2}= 0$

And I rearranged it but got nothing worth noting.

 

[Orodruin]

The coefficients in front of each power of x must cancel for the equation to hold. Also, expand about x=0 ...