Combining Power Series for Airy's Equation Solution

[David Laz]

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

<br /> \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.

 

[amcavoy]

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
<br /> \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?

 

[*melinda*]

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]

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.

 

[David Laz]

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