Say you have an un-damped harmonic oscillator (keep it simple) with a sine or cosine for the forcing function. We can exploit Euler's equation and solve for both possibilities (sine or cosine) at the same time. Then, once done, if the forcing function was cosine, we choose the real part as the particular solution. If it was sine, we choose the imaginary solution. Can someone say, in words, how we know this works? I know it works because I can do it. How do we justify it? Most books just do it, but never explain why it works. I am NOT looking for an advanced mathematical proof. I just want to know, in simple words, how we know this works.