Solution for a differential equation

Question

I am stuck in getting the solution for the following non-linear differential equation: \begin{equation*} x^2 + B\frac{dx}{dt} = A\sin(wt) \end{equation*} Is there any method to solve this kind of equation?

Answer 1

Substitute $x= Q \frac{u'}{u}$

$$Q^2\frac{u'^2}{u^2}+BQ'\frac{u'}{u}+BQ\frac{u''}{u}-BQ\frac{u'^2}{u^2}=A\sin (wt)$$

Pick $Q$ such that

$$Q^2\frac{u'^2}{u^2}-BQ\frac{u'^2}{u^2}=0$$

We get $Q=B$, $Q'=0$. Multiply everything by u:

$B^2u''=A \sin (wt) u$

Which, I'm afraid, has no solution in terms of known functions.

Answer 2

The general solution cannot be expressed in terms of a finite number of elementary functions. A closed form (see below) involves the Mathieu's functions : http://mathworld.wolfram.com/MathieuFunction.html

If those special functions are not implemented on your mathematical software, use numerical methods for solving the ODEs. And even if the Mathieu functions are avalaible, it is generally simpler to use numerical methods in pratice.