Help!! what would you do for a problem such as this?

Question

Show that the function f(x) equal to summation (0- infiniti) {[(-1)^n]x^(2n)}/(2n)!
is the solution of the differential equation f ''(x)+ f(x)=0.

help is greatly appreciated! thanks!

Answer 1

Well, simply find the second derivative of sum {[(-1)^n]x^(2n)}/(2n)!, and just remember that differentiation is a linear transformation, and so d/dx sum g(x,n) = sum d/dx g(x,n).

Once you have the second derivative, use it and the function f, add them together and show that it equals zero.

As far as working with sums, it helps to remember the properties of summations. Try index shifts, factoring out constants, etc....

Just be careful with index shifts....

This is how it goes:

sum[n=a..b] h(n) = sum[n=a-k..b-k] h(n+k)

And also, you may need to strip off terms in a sum to force limits to match....

sum[n=0..k] h(n) = h(0) + sum[n=1..k] h(n)

Answer 2

Write out the first few terms of the series. Then take the 2'nd derivative of each term and show that it's identical (except for sign) to the terms of the original series so they add up to zero.

Doug