I have the following differential equation: $\frac{d^{2} y}{d t^{2}}=-B \frac{d y}{d t}-C y$, with initial conditions: $\dot{y}(0)=0 \text { and } y(0)=1$, and values of B and C: $B=0.1 \text { and } C=1$.
I am given the analytic solution as $y(t)=A \exp \left(-\frac{t}{\tau}\right) \cos \left(\omega^{\prime} t+\delta\right)$
However, I have been asked to find $\tau \text { and } \omega^{\prime} \text { in terms of } B \text { and } C$, and I just don't understand how this is possible seeing as we have only one equation in B and C. We can find $\dot{y} $ and $ \ddot{y}$ in terms of $y$, $\tau \text { and } \omega^{\prime}$ but this gives us one equation linking B and C to $\tau \text { and } \omega^{\prime}$. I tried to use the initial conditions but this seemed to imply $C=0$. Is there something obvious I am missing here?