solution of a Fredholm equation of first kind

Question

I was recently studying about solution of the following homogeneous Fredholm equation of first kind $$x=\lambda\int_0^1e^{x-t}y(t)dt$$ if there's any . But the method of regularization (since the kernel is separable) and homotopic perturbation method both give a rather diverging solution for this problem . So I suspect if any solution exists at all . Is there any other method for any approximate solution to this kind of problem ? If this problem has indeed no solution , then what is the theoretical reason behind it ? Any help is appreciated .