I'm trying to face this nonlinear differential equation:
$$ y''(x)+\omega^2\sin\,y(x)=a\,x \,\;(1)$$
and I'm interested to found the solution of $ y'(x)$ (an first integral)
The homogeneous part of previous one ode is like a nonlinear free (non forced) pendulum diff. eq. : $$ \theta''+\omega^2\sin\theta=0$$ Then, the ode that i'm trying to solve it's similar to forced pendulum differential equations.
The first integral of homogeneous solution of (1) it's easy to solve : $$\frac{(y'(x))^2}{2}-\omega^2\cos \,y=\mathrm{const}$$
But, is there a solution for fist integral of (1) in a case of forcing of the type $f(x)=a\,x$ or otherwise?
For large $x$ the right side is large, while the $\omega^2 \sin(y)$ term is bounded. Thus it may be useful to consider this differential equation as a perturbation of $y'' = ax$. We can write $$ y(x) = \sum_{k=0}^\infty \omega^{2k} y_k(x)$$ where $$\eqalign{y_0(x) &= y(0) + y'(0) x + a x^3/6\cr y_1(x) &= - \int_0^x dt \int_0^t ds\; \sin(y_0(s))\cr y_2(x) &= - \int_0^x dt \int_0^t ds\; y_1(s) \cos(y_0(s))\cr \text{etc}}$$
I suspect that each $y_k$ for $k \ge 1$ will be asymptotic to some straight line as $x \to \infty$.