Discuss the convolution theorem to solve the ode

Question

Assignment:

Consider the ODE with the boundary conditions

y⁽⁴⁾ + βy = g(y) -∞ < x < ∞

y(x) bounded as |x| → ∞

Assume that β is real and positive and that g(x) behaves in such a way so that a bounded solution is possible.

(a) Compute the Fourier transform of the solution

(b) Use the convolution theorem to solve the ODE and express the solution as an integral involving g(x).