What kind of differential equation is this?

Question

What kind of differential equation is this?

$x^2*\frac{d^2y}{dx^2}+x*\frac{dy}{dx}+(x+1)*y=0$

It's close to an euler equation which is

$x^2*\frac{d^2y}{dx^2}+x*\frac{dy}{dx}+y=0$

but it's also close to Bessel equation which is

$x^2*\frac{d^2y}{dx^2}+x*\frac{dy}{dx}+(x^2+a^2)*y=0$

But does not fit either one right. Please let me know how to go about solving it?

Answer 1

This ODE belongs to the family of the Bessel equations. The solutions of the more general form of ODEs : $$x^2y''+(1-3p)xy'+(\lambda^2q^2x^{2q}+p^2-\nu^2q^2)y=0$$ are : $\qquad\qquad y(x)=c_1 x^p J_\nu(\lambda x^q)+c_2 x^p J_{-\nu}(\lambda x^q)$

In the present case : $p=0 \:;\: q=1/2 \:;\: \lambda=2 \:;\: \nu=\pm 2i $ $$y(x)=c_1 J_{2i}(2 \sqrt{x})+c_2 J_{-2i}(2 \sqrt{x})$$