# Expressing as a single sum.

I know this is pretty easy, but for this particular question I'm having difficulty.
its for the Power series solution of the DE y''+yx=0

$$\sum\limits_{n = 0}^\infty {(n - 1)nC_n } x^{n - 2} + \sum\limits_0^\infty {C_n } x^{n + 1}$$

This is ths sum I've come up with and I need to express it as a single sum. I can't seem to do it though. Any help will be greatly appreciated.

## Answers and Replies

David Laz said:
I know this is pretty easy, but for this particular question I'm having difficulty.
its for the Power series solution of the DE y''+yx=0
$$\sum\limits_{n = 0}^\infty {(n - 1)nC_n } x^{n - 2} + \sum\limits_0^\infty {C_n } x^{n + 1}$$
This is ths sum I've come up with and I need to express it as a single sum. I can't seem to do it though. Any help will be greatly appreciated.
Well, what do you know about the sum of series with the same indices?

The index of your first sum is not correct.

Remember that every time you take a derivative you loose a constant term.

After you correct your index you can then change it to something more desirable.

Tide
Science Advisor
Homework Helper
Incidentally, if you didn't know, your DE is just a variant Airy's Equation (with x replaced by -x) and represents waves propagating in a medium whose properties (index of refraction, water depth, etc.) vary linearly in space.

Excellent. Thanks for the help.

I believe we study Airy's equations in greater detail later on in the course. We touched on them briefly in my Quantum Physics class last semester though. :D