Second Order Non-Linear Ordinary Differential Equation

Question

I have the equation $$x_{tt}+cx_t+x=x^2$$ where $c$ is constant and $x=x(t)$.

If the $x^2$ wasn't on the right hand side of the equation then I could solve this easily by the method of characteristics. However since it is there, how do I deal with it? Is there a common transformation to use in this situation?

Answer 1

Let $p=x_t$ and consider $p$ as a function if $x$. Then $$ x_{tt}=\frac{dp}{dt}=p\,\frac{dp}{dx}. $$ The equation becomes $$ p\,\frac{dp}{dx}+c\,p+x=x^2. $$ This is a linear first order equation. Solve for $p$ as a function of $x$ and then for $x$ as a function of $t$. Unfortunately it does not seem to have an easy solution.

This procedure can be carried out whenever the independent variable ($t$ in this case) deos not appear explicitely.