Nonlinear transform can separate function composition?

[Yunjia]

I am solving a nonlinear ODE in the form of Newton's Second Law. In the equation, there is a Heaviside Theta Function of the function which I am solving ($\theta (x(t)$). Since it is quite troublesome to have both the left side of the ODE and the imput of the ODE to contain function of unknown function, I am considering using a transformation which can be nonlinear because linear transformation cannot help me separate the composition of two functions. Is there an analytical way to solve the equation?

P.S. Here is my equation
$$x''(t)+\omega_0^2 x(t)=[\vartheta(x(t)+b) \cdot \vartheta(x(t)-b)] \cdot \sin(\omega t)$$ where $\omega$ and $\omega_0$ are independent.

 

[Stephen Tashi]

there is a Heaviside Theta Function of the function

If $b > 0$ is $(\vartheta(x(t) + b) * \vartheta(x(t)-b)) = \vartheta(x(t) - b )$ ?