If you check journals online that uses Differential Transform Method, you will understand it. As for the transform of the differential equation you posted y'''(x)+xy''(x)-y(x)=0; 

(k+1)(k+2)(k+3)Y(k+3)+sum(delta(l)(k-l+1)(k-l+2)Y(k-l+1),l=0..k)-Y(k)=0.

You can still check Sobamowo et al, 2018 on the use of DTM

@badzline. O. Try to check your initial conditions. You are specifying too many points which is almost impossible to determine the solution.
If it is an IVP then, you specify at one point and two point for BVP.

You have an order 3 ODE, then you must have 3 specify conditions. Try to recheck your conditions.