I am trying to solve the following differential equation $$x''+sin(x)=f(t),$$ 1-I know the solution of the homogeneous system $x_h(t)$, is it possible to use it to find the general solution of the non-homogeneous system? I am looking for a solution in the following form $$x(t)=\int_{-\infty}^t \xi(t-t') f(t') dt' $$ but i don't know how to find $\xi (t-t')$.
2-Is it possible to find x(t) in terms of fourier series?
3- What other approximation methods can be used to solve this system? f(t) is a fluctuation force sampled from Gaussian distribution with zero mean and $\sigma=1$
The homogenous equation is the equation of the mathematical pendulum with general solution the Jacobi amplitude
$$x(t)= 2 \text{am}(\frac{\omega}{2} (t-t_0) | i \frac{2}{\omega}) , \quad \text{math notation}$$
$$x(t)= 2 \text{am}(\frac{\omega}{2} (t-t_0), - \frac{4}{\omega^2}) , \quad \text{technical notation}$$
Verification by $$\sin(2 \text {am}) = 2 \text{sn} * \text{cn}, \text{am}'=\text{dn} $$
The amplitude is either oscillating for$\omega <2$ or rotating for $\omega>2.$
Between both the solitary single loop for $\omega=2$, $\text{am}(t,1)$
$$ x(t)= 4 \ \tan^{-1}(e^{t-t_0})$$
The homogenous equation is solved by the energy method, multiplication by $x'=\frac{dx}{dt}$, solving for $dt$ yielding by simple integration the inverse function as an elliptic integral $F$
$$t-t_0 = \ \text{F}(x,k).$$
By nonlinearity and our knowlegde about the highly sophisticated methods to solve the sine-Gordon PDE, to find the perturbation solutions of the mathematical pendulum simply by a linear additive method of impulses $f(t)dt$ seems to be improbable or impossible.