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## Homework Statement

A particle of mass 'm' is subject to a spring force, '-kx', and also a driving force, F

_{d}*cos(w

_{d}t). But there is no damping force. Find the particular solution for x(t) by guessing x(t) = A*cos(w

_{d}t) + B*sin(w

_{d}t). If you write this in the form C*cos(w

_{d}t - [tex]\phi[/tex]), where C > 0, what are C and [tex]\phi[/tex]? Be careful about the phase (there are two cases to consider).

## Homework Equations

basic inhomogeneous second-order equation solving.

## The Attempt at a Solution

So I got the equation F = F

_{d}*cos(w

_{d}t) - k*x = m*x'', and put it in the form x'' + w

^{2}*x = F*cos(w

_{d}t), where w

^{2}[tex]\equiv[/tex] k/m, and F [tex]\equiv[/tex] F

_{d}/m. I took the first and second derivative of the equation they told us to use for our guess, and subbed them into x'' + w

^{2}*x = F*cos(w

_{d}t). I ended up with A(w

^{2}- w

_{d}

^{2})*cos(w

_{d}t) + B(w

^{2}- w

_{d}

^{2})*sin(w

_{d}t) = F*cos(w

_{d}t). I therefore deduced that F = A(w

^{2}- w

_{d}

^{2}). Therefore the particular solution would be x

_{p}(t) = A(w

^{2}- w

_{d}

^{2})*cos(w

_{d}t). Is there anyway that this can somehow simplify for the form they want us to put it in, C*cos(w

_{d}t - [tex]\phi[/tex]), or did I do something wrong?