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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.

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  • $\begingroup$ @LutzL That was my mistake. I’ve fixed it $\endgroup$ – Егор Пономарёв May 31 at 12:16
  • $\begingroup$ 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. $\endgroup$ – LutzL May 31 at 12:17
  • $\begingroup$ @LutzL Frankly speaking, it just should exist $\endgroup$ – Егор Пономарёв May 31 at 12:21
  • $\begingroup$ 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.) $\endgroup$ – LutzL May 31 at 12:24
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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".

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