Question

Given the ODE
y''+sin(x)*y=0
Find the first 4 nonzero terms in each of 2 power series solutions about x=0.

Answer 1

I am comfortable with finding series solutions to equations with terms whose expansions can be written in closed Riemann sum form. In this case, I have a problem with the sin(x)*y term.
I know it the first several terms of its expansion are:
\[\sin(x)*y = a_0 +x(a_0)+x^2(a_1)+x^3(a_2-a_0/3!)+x^4(a_4-a_1/3!)+...\]
But I'm not sure what to do next.