Solving a Volterra Integral Equation of the 2nd Kind

Question

Can anyone help me in finding a closed-form solution to the integral equation

$$x\left(t\right)=1-\lambda \int _{0}^{t}e^{-\alpha \left(t-\tau \right)} \cos ^{2} \left(k\tau \right) x\left(\tau \right)d\tau$$

Thanks a lot.

Answer 1

To solve this equation, first derive it :

As it is often the case for ODEs, the solution involves an integral which cannot be expressed in terms of a finite number of standard functions. In these cases, the integral is considered as a closed form itself and the given solution (expressed with the integral in it) is the final solution.

Also, this is the proof that any other method of solving will lead to the same result with an integral in it. So, it should be a waste of time to continue the search for other methods of solving and there is no hope for a simpler solution.

Note that, it doesn't exclude that simpler form might exist without integral in it, in case of particular values of the parameters $\alpha,k$, but not in the general case with any $\alpha,k$.