I like to use the series method, I am having problems constructing the solution to y''+ay=0, a>0
The solution should be y=Asin(a^.5x)+Bcos(a^.5x)
In the recursion relation I get -aCn/(n+2)(n+1) = Cn+2
However to construct sin, I need 'a' to have fractional power which I cannot seem to get.
i.e. for odd n, Cn should have 'a' with a fractional power like n+.5 or a^(n+.5) but I can only see a^n in Cn for all n. Ideally I like to deduce the correct form of the power of 'a' in the series method instead of writing it because I know the correct solution before hand using another method.