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Someone help me to solve this system\begin{align*} \begin{cases} \ddot{x}_1(s) -\alpha \, x_1(s) \,\ddot{x}_2(s) &= 0 \\ \\ \ddot{x}_2(s) -\alpha \, x_1(s) \,\ddot{x}_1(s) - \alpha \, \dot{x}_1^2(s) &= 0 \end{cases} \end{align*}

Thank you in advance

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  • $\begingroup$ from where Comes this System? $\endgroup$ – Dr. Sonnhard Graubner Oct 4 '17 at 18:01
  • $\begingroup$ Based on a calculation made for compute the Lagrangian of an unbounded operator on $\mathbb R^2$. $\endgroup$ – Z. Alfata Oct 4 '17 at 18:04
  • $\begingroup$ To start with, it is useful to notice that $x_2$ only appears as a second derivative. $\endgroup$ – Paul Oct 4 '17 at 18:28
  • $\begingroup$ @ Paul, if we replace $x_2$ by its expression in the first equation, the system remains somewhat complicated $\endgroup$ – Z. Alfata Oct 4 '17 at 18:42
  • $\begingroup$ $$x_1(s)=\frac{1}{\alpha},x_2(s)=C_1s+C_2$$ is one solution $\endgroup$ – Dr. Sonnhard Graubner Oct 4 '17 at 19:12

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