Solution of the second-order ODE with polynomial coefficients $(a-x)\,x\, y''(x)+\left(\frac{a}{2}-2 \,x\right)y'(x)+\lambda \,x^2y(x)=0$.

Question

I am searching for solutions of the following second order differential equation:

$$(a-x)\,x\, y''(x)+\left(\frac{a}{2}-2 \,x\right)y'(x)+\lambda \,x^2y(x)=0,\qquad (0<x<a)$$ with $y(a)=0$ and $y'(0)=0$.

As far as I know, there is no finite polynomial solution. In a book I have found a solution when the factor $x^2$ in the last term is replaced by 1.

EDIT: Boundedness of $y(x)$ for $x\to 0$ is an alternative condition (instead of $y'(0)=0$).