# How do I solve a non-linear first order differential equation?

I have the specific first order non-linear differential equation as shown below:

$$\frac{d\Omega}{d\theta} - M(\theta)\frac{1}{I\Omega} = \frac{D}{I}$$

Where D and I are constants. And $M(\theta) = A\cdot\sin(2\theta) + B$, where A and B are constants. Could anyone advice me if this is solvable, and if so, what are the steps I should take?

I have two news for you. :-)

The first news is good. This equation is reducible to a known equation and your are lucky, because few hours ago Player100 me a hint for an other question, which helped me with yours too. Nevertheless, domains of my math competence are far from differential equations, so I can miss some subtleties.

If $D=0$ then substitute $x=\theta$, $y=\Omega$ and obtain

$$y’y=\frac {A\sin 2x+B}I$$

$$(y^2)’=\frac {A\sin 2x+B}{2I}$$

$$y^2=\int\frac {A\sin 2x+B}{2I}dx=\frac{2Bx-A\cos 2x}{4I}+C,$$

where $C$ is an arbitrary constant.

If $D\ne 0$ then substitute $x=\theta$, $y=I\Omega/D$ and obtain

$$y’y-y=\frac I{D^2}(A\sin 2x+B).$$

This is Abel equation of the second kind (see [PZ, 1.3.1]). There is written “Given below in this section are all solvable Abel equations known so far”. And here comes the bad second news. This equation is not listed there (possible, except Equation 76, which doesn’t look applicable). So I’m afraid your have to look for a solution in a form of a series $$y(x)=y(x_0)+\sum_{n=1}^\infty a_n(x-x_0)^n.$$

References

[PZ] Andrei D. Polyanin, Valentin F. Zaitsev. Handbook of exact solutions for ordinary differential equations, second edition, Chapman and Hall/CRC, 2003. (available at LibGen).