what type of differential equation is it?

Question

someone can indicate me that type of equation is and its solution method.

$$x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}+4=0$$

I understand that it performs the second order

Answer 1

It is a second order homogeneous differential equation with non-constant coefficient.

Answer 2

The homogeneous part is an Euler-Cauchy DE, so it is easy to find basis solutions.

For instance by substituting $x=e^t$ to get an equation with constant coefficients.

Or by computing a power series, Frobenius' method.

---

Or you can see that it is a first order ODE with the usual solution formula after substituting $u=y'$.

---

Or you can combine the terms to the easily integrable form $$x(xy')'=-4$$

Answer 3

Substituting $$\frac{d^2y(x)}{d^2x}=\frac{dv(x)}{dx}$$ and $$\mu(x)=e^{\int\frac{1}{x}dx}=x$$ we get $$\frac{d}{dx}\left(xv(x)\right)=-\frac{4}{x}$$ Can you finish?