TITLE: Unsolvable first order nonlinear differential equation? $\frac{dx(t)}{dt} + \sin(x(t)) = \sin(\omega t)$
QUESTION [8 upvotes]: I do want to solve the following differential equation analytically:
$$\frac{dx(t)}{dt} + \sin(x(t)) = \sin(\omega t)$$
I tried several methods to solve this equation, unfortunately without any success. In the last week, I have read a lot of papers related to that kind of prototype and have looked into all the books that deal with first order differential equations. It still seems to be impossible. Neither Wolfram-alpha, nor Matlab's symbolic toolbox can give me a solution for it. Because of that, I am wondering if there is actually a solution for that kind of differential equation?
Thanks for your help!

REPLY [11 votes]: Of course solutions do exist, we might just be unable to find closed form representations for them. This doesn't mean we can't compute them numerically: here are some IVP trajectories with $\omega=1$

You can plot just a fundamental patch $t\in[0,2\pi]$, $x\in[-\pi,\pi]$ thanks to the periodicity (I've highlighted in red the stable periodic solution and in blue the unstable periodic solution):

Alternative visualization:

Edit. Code for the plots, in Mathematica:
sol = ParametricNDSolve[{
    x'[t] + Sin[x[t]] == Sin[t],
    x[0] == x0},
   x, {t, 0, 4 \[Pi]}, {{x0, -3 \[Pi], 3 \[Pi]}}];
Plot[Evaluate@Table[
   x[x0][t] /. sol, {x0, -3 \[Pi], 2.5 \[Pi], .25}], {t, 0, 3.5 \[Pi]}]

per = NDSolve[{
    x'[t] + Sin[x[t]] == Sin[t],
    x[0] == x[2 \[Pi]]},
   x, {t, 0, 2 \[Pi]}];
Show[StreamPlot[{1, Sin[t] - Sin[x]}, {t, 0, 2 \[Pi]}, {x, -\[Pi], \[Pi]}],
 Plot[Evaluate[x[t] /. per], {t, 0, 2 \[Pi]}, PlotStyle -> Red]]