 Solution

Before the impulse the zero initial data force $y_2=0$. An impulse modeled by the Dirac delta function requires continuity of $y_2$: a jump in $y_2$ would introduce an unwanted derivative of a delta in $y_2^{\prime \prime }$. Integrating the equation across $t=a$ then gives the velocity jump $y_2^{\prime }(a+)-y_2^{\prime }(a-)=1$.

The causal Green function of a damped oscillator isThe jump conditions give the causal impulse responseHere $H$ is the Heaviside step function. For $\omega =0$, interpret the quotient continuously as $t-a$, giving $H(t-a)(t-a)e^{-k(t-a)}$. Both initial values are zero because $a>0$.