Second-Order Linear Equations · Challenge problem

The function $y_1 = e^{-2t}\cos 3t$ solves $y'' + by' + cy = 0$ for certain constants $b, c$. Another solution $y_2$ of the same equation has $y_2(0) = 0$, and the Wronskian $W = y_1y_2' - y_2y_1'$ equals $6$ at $t = 0$. Find $y_2\left(\dfrac{\pi}{6}\right)$.