I am trying to solve this system of differential equations: $$ \left\{ \begin{array}{c} \dot{x}_1 = 2t(x_1+x_2+t^2) \\ \dot{x}_2 = t(x_1^2-t^4-2t^2-1)+x_2^2 \end{array} \right. $$
The way to do it is to getone regular (easily solvable) differential equation from this system. Please, provide any ideas how to do it.
If you want to force it, isolate $x_2$ from the first equation and insert it and its derivative in the second equation. \begin{align} x_2&=\frac{\dot x_1}{2t}-x_1-t^2\\[1.5em] \dot x_2&=\frac{\ddot x_1}{2t}-\frac{\dot x_1}{2t^2}-\dot x_1-2t\\ &=t(x_1^2−t^4−2t^2−1)+\left(\frac{\dot x_1}{2t}-x_1-t^2\right)^2 \end{align} The last equality is a second order ODE in $x_1$ only, however I would not call that "simple".