$ty''+4y'=0$ , how to solve it by reduction?

Question

I'am struggling with this equation. I need to use reduction method, where $u=y(t)$ and $y''(t)=u'u$. What I get is: $u'tu+2u=0$, then I divide it by $tu$ and get $u'+\frac{2}{t}=0$ (of course assumed $ut$ is not $0$). But what now? Integrate it? In respect to what? What type of equation is it?

Answer 1

$$ty''+4y'=0$$

Let $$y'=u$$

$$y'' =u'$$

$$tu'+4u=0$$ $$u'=-(4/t) u$$

$$u=ct^{-4}$$ $$y'=ct^{-4}$$ $$y=c_1t^{-3}+c_2$$