The solution of the second order DE...

(1)

... is supposed to be analytic in x=0 so that we can write...

(2)

The computation of the derivatives in x=0 is performed as follows. The initial conditions give us and . From (1) we obtain directly...

(3)

Deriving (3) we obtain...

(4)

The computation of the successive derivatives is left to You...

Kind regards