solving third order nonlinear ordinary differential equation.

Question

Can anybody give any hints how to solve any kind of particular solution involving at least one arbitrary constant of this ode: $p^2y^{\prime \prime \prime}+ yy^{\prime}+y+ax+b=0$,

where $a$,$b$,$p$ are constants.

Answer 1

Ansatz: $y=cx+d$, $y'=c$, $y'''=0$. $$0=p^2y'''+yy'+y+ax+b=(cx+d)c+cx+d+ax+b= (c^2+c+a)x+(cd+d+b).$$ Conclude: $y$ solves your ODE if: $$c=\frac{-1\pm\sqrt{1-4a}}{2},$$ $$d=-\frac{b}{c+1}.$$

So there's two solutions. I do not know if there are others.