 # I have the specific first order non-linear differential equation as shown below: boloman0z 2022-06-25 Answered
I have the specific first order non-linear differential equation as shown below:
$\frac{d\mathrm{\Omega }}{d\theta }-M\left(\theta \right)\frac{1}{I\mathrm{\Omega }}=\frac{D}{I}$
Where D and I are constants. And $M\left(\theta \right)=A\cdot \mathrm{sin}\left(2\theta \right)+B$, where A and B are constants. Could anyone advice me if this is solvable, and if so, what are the steps I should take?
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If D=0 then substitute $x=\theta$, $y=\mathrm{\Omega }$ and obtain
${y}^{\prime }y=\frac{A\mathrm{sin}2x+B}{I}$
$\left({y}^{2}{\right)}^{\prime }=\frac{A\mathrm{sin}2x+B}{2I}$
${y}^{2}=\int \frac{A\mathrm{sin}2x+B}{2I}dx=\frac{2Bx-A\mathrm{cos}2x}{4I}+C,$
where C is an arbitrary constant.
If $D\ne 0$ then substitute $x=\theta$, $y=I\mathrm{\Omega }/D$ and obtain
${y}^{\prime }y-y=\frac{I}{{D}^{2}}\left(A\mathrm{sin}2x+B\right).$
This is Abel equation of the second kind
$y\left(x\right)=y\left({x}_{0}\right)+\sum _{n=1}^{\mathrm{\infty }}{a}_{n}\left(x-{x}_{0}{\right)}^{n}.$

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