Differential Equation second order using series (reccurrence relation)

Question

(dy^2/d^2x)+(1-x)(dy/dx)-y=0

Initial Conditions y(0)=-1; dy/dx(0)=2

I am able to solve the reccurence relation, but Im not sure how to find the values of a0 and a1 using the inital conditions. Can someone help please.

Answer 1

assuming that you are dealing with the series y= a_o + a₁x +a₂x² + a₃x³ + ...............

then y(o) = a_o and from your initial conditions, y(0) = -1 so, a_o = -1

when you differentiate y'(x) = a₁ + 2a₂x + 3a₃x² + ............................

then y'(0) = a₁ and from your initial conditions y'(0) = 2 so, a₁ = 2