Power series - math problem?

Question

Find a power series solution for each of the initial value problems below:

(a) y' (x) = cos x^2, y(0) = 0

(b) y'' - xy=0, y(0)=1, y' (0) = 0

Thanks for any help!

Answer 1

a) write a series for cos t {you should know this series}, replace "t" with "x^2", integrate term by term.....b) y(0) = 1 implies a[sub 0] = 1 , y '(0) = 0 implies a[sub 1] = 0. Try the series....sigma [0 to infinity] a[sub n] x^n for y(x). You should find the recursive equation a[sub (n+3)] = [a(sub n)] /[(n+3)(n+2)].....compute a few of the terms until you get a "feel" for the [3n]th term...it can be written {check the work} as 4*7*10*...[3n-2] divided by [3n]!...hope this helps as YOU do the problem.

Answer 2

So what is your problem? Is it that you don't understand the power series method for solving differential equations? That you understand the method but have run into problems applying it in these cases?

If the latter, I suggest you show us what you've already done. In any case, perhaps you should update the post with more information about what sort of help you need.