Question

separation of variables dy/dx= (1+t/1+y)^2. the whole term is raised to the second power, no parentheses on the inside terms

Answer 1

Where did the t come from? Seems a bit strange.

Answer 2

\[\frac{dy}{dt}=\left(\frac{1+t}{1+y}\right)^2~?\]
Separating variables yields
\[(1+y)^2~dy=(1+t)^2~dt\]
Make substitutions: \(u=1+y~\Rightarrow~du=dy\) and \(v=1+t~\Rightarrow~dv=dt\).
\[\int u^2~du=\int v^2~dv\\
\frac{1}{3}u^3=\frac{1}{3}v^3+C\\
~~~~~~~~~~~~~\vdots\]

Answer 3

@SithsAndGiggles : That is what I was gonna write, and I am almost positive that OP made a mistake in the question. Nonetheless, OP, if you reading this, confirm this solution.