Problem 1d correction, hint

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For part (d) of problem 1 of Hw#2, every instance of $t$ should be replaced by $\tau$, so the wording should be changed to read: "Consider the case where $\zeta =0$ and $v(\tau )=\sin \omega \tau ,\omega >1$. Solve for $z(\tau )$, the normalized output of the oscillator, with initial conditions $z_{1}(0)=z_{2}(0)=0$."
If you've already solved it using $t$ instead of $\tau$ you will get equal credit (it is just a little bit more complex).
To solve this problem, you can use the "method of undetermined coefficients" (see, for example, http://www.efunda.com/math/ode/linearode_undeterminedcoeff.cfm) to solve for the steady-state frequency response solution. Then you can add to it a homogeneous solution that cancels the initial condition from the steady state so that the given initial conditions are satisfied. i.e., find $z_{{homog.}}$ from $z=z_{{homog.}}+z_{{partic.}}$.