What would be the general homogeneous solution for x(t)?View attachment 36773
If anyone can recommend or suggest anything, that would be great. Any website or video I can go to and learn to solve this would be awesome too.
I might be missing something, but I wouldn't expect this to work: Fourier series for [imath]f(t)[/imath] will have infinite number of terms, so we'd end up with an infinite number of unknowns. The solution I found is, IMHO, simpler than that.For the particular integral, begin by calculating the Fourier series for f(t), it will be a series of cosine terms since f(t) is an even function.
Assume a matching series for x(t), same cosine terms but unknown coefficients.
Substitute into the de and equate coefficients across the equation. That should get you the unknown coefficients in the x(t) series.
For the general solution of the equation add that to the solution of the homogeneous equation.
The question asks for the solution as a series, (assumed infinite in length).I might be missing something, but I wouldn't expect this to work: Fourier series for [imath]f(t)[/imath] will have infinite number of terms, so we'd end up with an infinite number of unknowns. The solution I found is, IMHO, simpler than that.
You are right, I missed that part.The question asks for the solution as a series, (assumed infinite in length).
The series for x(t) will have an infinite number of terms but so does the Fourier series on which it is based.
The relationship between the two sets of coefficients is fairly simple making it easy to write down the general term of the x(t) series.