Solve DE w/ Power Law Trick | L^2/(y^3)

[Anito]

Homework Statement

I have no idea how to solve this differential equation:(d^2y/ds^2)=L^2/(y^3)

where L is constant. It looks like a inhomogenius DE but what should I do with y^3?

 

[Dick]

Try looking for a power law solution y=As^k.

 

[HallsofIvy]

Dick, it that were $y^2$ on the right side that would work (it would be an "Euler-type" equation) but I don't think it works here. Since the independent variable, s, does not appear explicitely, I would try "quadrature".

Let v= dy/dt so $d^2y/dt^2= dv/dt= (dv/dy)(dy/dt)= v dv/dy$. The equation becomes v dv/dy= L/y³. vdv= Ly^-3dy. Integrate that to get (1/2)v²= (-L/2)y^-2+ C. Since v= dy/dt, that is
$$\frac{dy}{dt}= \sqrt{C- Ly^{-2}}$$

 

[Dick]

Thanks, Halls. The power law trick does give you a particular solution proportional to s^(1/2), but that way you get a more general solution.