Differential Equation (cosx)y +y=0 using power/taylor series

[cybla]

Differential Equation (cosx)y"+y=0 using power/taylor series

Homework Statement

Find the first three nonzero terms in each of two linearly indepdent solutions of the form y=\sum(c_{n}x^{n}). Substitute known Taylor series for the analytic functions and retain enough terms to compute the necessary coefficients.(cosx)y"+y=0

Homework Equations

The Attempt at a Solution

I substituted the taylor series for cosx, and the general power series into the equation, to get...

\sum(\frac{(-1)^{n}x^{2n}}{(2n)!})\sum(n(n-1)c_{n}x^{n-2})+\sum(c_{n}x^{n})=0

I then multiplied the first term out by series multiplication but that did not seem to get me anywhere. I am stuck at this part

By the way... for the second series (second derivative) the index begins at n=2 and the other two are n=0...sorry i do not know how to put the subscripts on the series

 

[ehild]

Use different summation indices in the different sums and collect the terms containing xⁿ from the product.

ehild

 

[nickalh]

LaTeX or Math Symbol Notation
On my web browser, there's a sigma/sum/ \sum sign at the right end of the row with, Bold, Italic, underline. Click it to open a whole slew of notations. This box lists \sum on the operator submenu. I personally, use the Subscript and Superscript menu options. One can also right click on someone's notation to "show source" and see how individual code works. Somewhere, there's a thread for the sole purpose of explaining the notation. Finally, LaTeX is standardized, so in theory it is possible to download a LaTeX editor to create the markup code, then copy and paste into here.

The lack of clarity for index variable, initial value, and final value is throwing me.

\sum ^{i=0}_{n} or <br /> \sum ^{n=1}_{m}


And yes, IIRC, ehild has the right idea. For the 2nd derivative series, substitute,
n_{a}=n_{b}\: -2
so that all sums have the same initial value.

However, I don't remember, is it ordered on highest degree first or lowest degree first?
For the cos x series,
x^(2n), x^(2n-2), or x^0, x^2, x^4?

 

[cybla]

So if i change the summation index for the second derivative i get...

\sum^{\infty}_{n=0}(n+2)(n+1)c_{n+2}x^{n}

and the first power series reamins the same. Also for cosx it is


\sum^{\infty}_{n=0}\frac{(-1)^{n}(x)^{2n}}{(2n)!}



However, once i get to this stage... do i multiply out the cosx and y"? Or should i simply just take n=0,n=1,n=2...etc. of the cosx and y" and multply those out and compare coefficients to solve for the c's?

 

[cybla]

I figured it out. Thank you for your help