How I can solve a system of this type

Question

How I can solve a system of this type $$\begin{cases} \ddot{x_{1}}(s) - x_{1}(s) \, \dot{x_{1}}^{2}(s) &=0\\ \ddot{x_{2}}(s) - x_{1}(s) \, \ddot{x_{1}}(s) + \dot{x_{1}}^{2}(s) &=0 \end{cases}$$ Thank you in advance

Answer 1

Note that first equation does not depend on $x_2$. Hence, you can solve it first, then substitute the solution to the second equation, and then solve second equation.

First equation can be solved by substitution $x_1' = p(x_1)$, where $p$ is unknown function. Then we have $x_1'' = \frac{d}{ds}p(x_1(s)) = p' p$. Therefore, you first equation reduces to $$ p'(x_1) p(x_1)=x_1 p^2(x_1). $$ Hence, either $p(x_1) = 0$, or $p'(x_1)=x_1 p(x_1)$. Both these equations are integrable, and you can find $p$ explicitly. Then, you integrate the equation $x'=p(x)$, and find explicitly $x_1(s)$.