# Math Help - [SOLVED] First order non linear non homogeneous DE

1. ## [SOLVED] First order non linear non homogeneous DE

I must solve the following DE: $x'(x-x^2)=t+t^2$.
My attempt: $(x-x^2)dx=(t+t^2)dt \Rightarrow x^2 \left ( \frac{1}{2}-\frac{x}{3} \right)= t^2 \left ( \frac{1}{2}+\frac{t}{3} \right ) +C$. I feel I'm on the wrong direction. How would you tackle it?

2. In my opinion You are proceeding in the right direction!... what You have to do now is to find an explicit expression of x as function of t by resolving the algebraic equation...

$\frac{x^{2}}{2} - \frac{x^{3}}{3} - \frac{t^{2}}{2} - \frac{t^{3}}{3} + c = 0$ (1)

The (1) is a third order equation so that You obtain [in general...] three different solution and among them the 'right' solution is determined [when possible...] by the 'initial conditions'...

Kind regards

$\chi$ $\sigma$