# How can solve this nonlinear system?

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*}

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