I am trying to solve a recurrence relation using generating functions method. After some long calculations, I have arrived to this second-order differential equation: $ 0.5 x^5 y”(x)+(2x^4+x^3)y'(x)+\left(x^3+x^2+x-1\right)y(x)+1=0$

and these conditions: $ y(0)=1, y'(0)=1$ . $ y(x)$ is the function that needs to be expanded as Taylor Series at $ x=0$ to obtain the sequence from the coefficients. However, when I try to solve it both using DSolve and NDSolve, I have no luck. With DSolve it just returns the request itself:

$ $ \text{DSolve}\left[\left\{0.5 x^5 y”(x)+(2. x+1) x^3 y'(x)+\left(1. x^3+x^2+x-1\right)y(x)+1=0,y(0)=1,y'(0)=1\right\},y,x\right]$ $

And with NDSolve I just receive errors and no equation:

`Power::infy: Infinite expression 1/0.^5 encountered. Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered. NDSolve::ndnum: Encountered non-numerical value for a derivative at x == 0.`. `

$ \text{NDSolve}\left[\left\{0.5 x^5 y”(x)+(2. x+1) x^3 y'(x)+\left(1. x^3+x^2+x-1\right)y(x)+1=0,y(0)=1,y'(0)=1\right\},y,\{x,0,1\}\right]$

How could I resolve this problem?