A particle of mass 'm' is subject to a spring force, '-kx', and also a driving force, Fd*cos(wdt). But there is no damping force. Find the particular solution for x(t) by guessing x(t) = A*cos(wdt) + B*sin(wdt). If you write this in the form C*cos(wdt - [tex]\phi[/tex]), where C > 0, what are C and [tex]\phi[/tex]? Be careful about the phase (there are two cases to consider).
basic inhomogeneous second-order equation solving.
The Attempt at a Solution
So I got the equation F = Fd*cos(wdt) - k*x = m*x'', and put it in the form x'' + w2*x = F*cos(wdt), where w2[tex]\equiv[/tex] k/m, and F [tex]\equiv[/tex] Fd/m. I took the first and second derivative of the equation they told us to use for our guess, and subbed them into x'' + w2*x = F*cos(wdt). I ended up with A(w2 - wd2)*cos(wdt) + B(w2 - wd2)*sin(wdt) = F*cos(wdt). I therefore deduced that F = A(w2 - wd2). Therefore the particular solution would be xp(t) = A(w2 - wd2)*cos(wdt). Is there anyway that this can somehow simplify for the form they want us to put it in, C*cos(wdt - [tex]\phi[/tex]), or did I do something wrong?