Bernoulli differential equation by using substitution
3(1+t^2) dy/dx = 2ty(y^3 - 1)
Plz help. Will greatly appreciate the detailed answer.

Question

Bernoulli differential equation by using substitution 
3(1+t^2) dy/dx = 2ty(y^3 - 1)
Plz help. Will greatly appreciate the detailed answer.

anonymous · Answer

$$\begin{align*}
3(1+t^2)\frac{dy}{dt}&=2ty(y^3-1)\\
3(1+t^2)\frac{dy}{dt}&=2ty^4-2ty\\
3(1+t^2)y^{-4}\frac{dy}{dt}&=2t-2ty^{-3}
\end{align*}$$
Set $u=y^{-3}$, then $\dfrac{du}{dt}=-3y^{-4}$. Then
$$\begin{align*}
-(1+t^2)\frac{du}{dt}&=2t-2tu\\
\frac{du}{dt}-\frac{2t}{1+t^2}u&=\frac{2t}{1+t^2}
\end{align*}$$
The equation is now linear in $u$.