Solve $a\frac{d^3y}{dx^3}+by\frac{d^2y}{dx^2}+c\frac{dy}{dx}=0$

Question

Is there any specific method of solving nonlinear differential equations? I want to solve the following differential $$a\frac{d^3y}{dx^3}+by\frac{d^2y}{dx^2}+c\frac{dy}{dx}=0$$ by the method of undetermined coefficients, but getting a term like $am^3e^{mx} + bm^2e^{2mx} + cme^{mx} = 0$. Please give some suggestions to solve.

Answer 1

$$a\frac{d^3y}{dx^3}+by\frac{d^2y}{dx^2}+c\frac{dy}{dx}=0$$ The DE is not linear so you can't use usual tricks for linear DE. Maybe you can reduce the order of the de. Substitute: $$\dfrac {dy}{dx}=p \text { and } p'=\dfrac {dp}{dy}$$= $$ap\dfrac d{dy}(pp')+bypp'+cp=0$$ $$ap(p'^2+pp'')+bypp'+cp=0$$ Note that $y=C$ is a solution of the original DE. Then: $$a(p'^2+pp'')+byp'+c=0$$