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:
d Ω d θ M ( θ ) 1 I Ω = D I
Where D and I are constants. And M ( θ ) = A sin ( 2 θ ) + 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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Answers (1)

Aiden Norman
Answered 2022-06-26 Author has 16 answers
If D=0 then substitute x = θ, y = Ω and obtain
y y = A sin 2 x + B I
( y 2 ) = A sin 2 x + B 2 I
y 2 = A sin 2 x + B 2 I d x = 2 B x A cos 2 x 4 I + C ,
where C is an arbitrary constant.
If D 0 then substitute x = θ, y = I Ω / D and obtain
y y y = I D 2 ( A sin 2 x + B ) .
This is Abel equation of the second kind
y ( x ) = y ( x 0 ) + n = 1 a n ( x x 0 ) n .

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