Infinite string wave equation

Infinite string at rest for t<0, has instantaneous transverse blow at t=0 which gives initial velocity of $\displaystyle V \delta ( x - x_{0} ) $ for a constant V. Derive the position of string for later time.

I thought that this would be $\displaystyle y_{tt} = c^{2} y_{xx} $ with $\displaystyle y_{t} (x, 0) = V \delta ( x - x_{0} ) $, and $\displaystyle y(x,0) = 0 $. So use the d'Alembert solution $\displaystyle y = f(x + ct) + g(x - ct) $. Then applying these gets the forms of f, g.

Is this correct? I am a bit nervous because I am self-teaching some of this wave equation stuff and I am not good at applying the theory to a practical question