# Get regular differential equation from a system

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.

• @LutzL That was my mistake. I’ve fixed it – Егор Пономарёв May 31 at 12:16
• Why do you suspect that such a radical simplification of your system exists? The system is non-trivially coupled, non-linear and non-autonomous. In general it is not possible to reduce this to one dimension. – LutzL May 31 at 12:17
• @LutzL Frankly speaking, it just should exist – Егор Пономарёв May 31 at 12:21
• Only if there is some constructive reason for it, like a physical symmetry. (I mean both that a one-dimensional reduction exists and that the resulting ODE is simple to solve.) – LutzL May 31 at 12:24

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".