How Do You Solve Nonautonomous Nonlinear ODEs Like dy/dt = A*y^n + g(t)?

[blumfeld0]

I need some help with a differential equation:

dy/dt = A*y^n + g(t)

where A and n are constants that can be any real numbers and g(t) is just
some function of t (e.g., g(t) = exp(-t/tau) or g(t) = constant or g(t) =
0). I think this is a nonlinear ODE.
If g(t) is a constant then it is an "autonomous" equation but if g(t) depends explicitly on t then it is "nonautonomous".
I can solve it if g(t) = 0. and g(t) = constant.
I can't find a solution if g(t) is some arbitrary function of t.

Any help would be much appreciated!

 

[HallsofIvy]

Yes, because of the "y^n" that's a non-linear equation. There is no general way of solving such equations. The solution will depend strongly on exactly what g(t) is.

 

[arildno]

Note for example if you choose A=1, n=2, g(t)=1, then your solution will blow up at a FINITE time. Other choices won't have this "pathological" behaviour, so the precise shape of g is crucial for saying anything particular about the solution of your diff.eq.