Help solving the initial value problem dy/dt = 2t/(y-t)

Question

I am trying to solve the following initial value problem for this differential equation:

$$\frac{dy}{dt} = \frac{2t}{y-t} , \hspace{20px} y(1) = 0$$

I have tried the substitution $v=\frac{y}{t}$ which transforms the ODE into a separable equation, but I don't think it was the best approach to solve this problem. Can you help me figuring out what to do?

Thanks!

Answer 1

$$\frac{dy}{dt} = \frac{2t}{y-t} , \hspace{20px} y(1) = 0$$ Substitute $y=vt$: $$v't+v = \frac{2}{v-1} $$ $$v't = \frac{2-v^2+v}{v-1} $$ This differential equation is separable.