# Solving 2nd-order ODEs

REF per request:

Title: Fourier Series: The origin of all we’ll learn
Author: Paul Loya


which I found very good to explain the origin of the Fourier series. However, I have a problem with 2 key points of the paper which are left unexplained. I was wondering if someone could put me on the right track:

In page 12, he goes on writing:

$$F''(x) = -m^2 F(x)$$

And then "we see that"

$$F(x) = A \cos(mx) + B \sin(mx)$$

This is the part I have a problem with. What's the method to get to the second equation. Why would $F(x)$ equals to such thing? Where is that coming from?

Later for the second part of the equation he uses he states:

$$G''(y) = m^2 G(y)$$

And "From elementary differential equations, we know that"

$$G(y) = Ae^{my} + Be^{-my}.$$

I didn't really go that far at school, so probably miss the bit where this was explained. I don't necessarily ask someone to explain me in detail something that might be taking a long time, but maybe just pointers to documents, wiki articles, name of these identities which I could lookup on the web myself... having that would already put me on the right path.

The second equation is just a solution of the first one, which is a second order differential equation

There are various methods of solving them, one is by simply 'guessing' the solution.

You could use the second equation

$$F(x) = A \cos(mx) + B \sin(mx)$$

and differentiate it twice by x

$$\frac{d^2 F(x)}{dx^2}=-Am^2\cos(mx)-Bm^2\sin(mx)=$$ $$=-m^2(A\cos(mx)+B\sin(mx))=-m^2F(x)$$

So you see that that solution is valid solution. ($F''(x)$ is just a short hand notation for $\frac{d^2F(x)}{dx^2}$)

The same can be done for second one.

You should download some book about differential equations prior to tackling the Fourier series. Schaum has good books, and there are lot of PDF's lying around the web ;)

• This is very good answer. Thank you so much. I really appreciate it. Can you recommend a book in particular which you find is good? Thank you again. Much appreciated. Commented Oct 28, 2013 at 9:07
• Is that the one you are suggesting? Schaum's Outline of Differential Equations, 3ed (Schaum's Outline Series) Commented Oct 28, 2013 at 9:08
• Yep, that's the one. You could also see about Arfken and Weber (scribd.com/doc/84183760/…), a great book about mathematical methods in physics. It has a chapter on DE there :) Commented Oct 28, 2013 at 9:16
• Glad I could help :) Commented Oct 28, 2013 at 9:23