Regarding solving ODE by Fourier transform, I read a nice reply by O.L.. After applying Fourier transform to an ODE to obtain an algebraic equation, the reply showed that some terms involving the delta function will show up during solving the algebraic equation, but also disappear for some reason unknown to me. More specifically, the reply said that the terms involving the delta function "represent the general solution of the homogeneous equation, so in our case $~α=β=0~$." I don't understand why "$~α=β=0~$"?
I don't find out how the initial conditions are used in the method by Fourier transform. In the solution by the method of Fourier transform, are there some undetermined coefficients to be determined from the initial conditions?
- How does the method of Fourier transform solve ODEs, in general?
- What are the advantages and disadvantages of solving ODEs by Laplace transform and solving ODEs by Fourier transform?
Thanks!
