Laplace transform of v(t)² for vehicle motion equation

[keebler71]

Ok...so this isn't homework (well past that at 39!) but I'm trying to brush off the old cobwebs and solve a vehicle motion problem. Specically, I'm trying to find the

Homework Statement

Ok...so this isn't homework (well past that at 39!) but I'm trying to brush off the old cobwebs and solve a vehicle motion dynamics problem. Specifically I’m trying to solve for the velocity history of the vehicle subject to accelerations that vary with the velocity and velocity squared. Eventually I’ll add a forcing term and I’d like to solve this using the Laplace Transform.

Homework Equations

Here is a generalized version of the equation:

v’ (t) = k1*v(t)^2 + k2^v(t) + k3

The Attempt at a Solution

The left side is easy:
L[v’(t)] = sV(s) - v(0)

But I’ve been stumped by the right side. Specifically, I am having trouble finding the L[v(t)^2] term. I’ve tried substituting into the definition of the LT and integrating by parts but not luck so far and can't find an similar example on the web or in a table... Any suggestions?

 

[vela]

I don't think the Laplace transform is appropriate for this problem because you don't have a linear differential equation due to the v² term. The differential equation is separable, though.