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?


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.

  • $\begingroup$ The presence of the term $p(dp/dx)$ forces me to ask: in what sense is this equation linear? I'm pretty sure that, in general, for solutions $p_1$, $p_2$, $a p_1 + b p_2$, $a, b \in \Bbb R$ is not a solution. Or do I miss something? $\endgroup$ – Robert Lewis May 4 '15 at 22:17
  • $\begingroup$ Indeed, this new form doesn't seem to be solvable by using an integrating factor? $\endgroup$ – misterE May 4 '15 at 22:19
  • $\begingroup$ Indeed it is not linear. Thanks for noting it. $\endgroup$ – Julián Aguirre May 5 '15 at 10:12

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