Your solution seems to be 'all right'!...A simple approach however for finding a solution of the ODE ...

(1)

... is to search for a search of the coefficients of the Taylor expansion...

(2)

The initial conditions give us and . The other derivatives are derived from (1)...

(3)

(4)

(5)

(6)

(7)

Now You are able to write the Taylor expasion till to the order 6...

(8)

Of course You can proceed to higher orders...

Kind regards