Laplace Transform, impulse response

Suppose a differential equation, $a\ddot{w}+b\dot{w}+cw=\delta(t)$, has the following unit impulse response:

What would its response to the unit step function be?

Using convolution we get

So, $v(t)=\left[-e^{-\tau}+\frac{1}{3}e^{-3\tau}\right]_0^t=-e^{-t}+1+\frac{1}{3}e^{-3t}-\frac{1}{3}=\frac{2}{3}-e^{-t}+\frac{1}{3}e^{-3t}$

Alternatively, we know that $w(t)=\mathop{\mathscr{L^{-1}}}\left\{\frac{1}{as^2+bs+c}\right\}$.

Comparing that to $\mathop{\mathscr{L}}\left\{w(t)\right\}=\mathop{\mathscr{L}}\left\{e^{-t}-e^{-3t}\right\}=\frac{1}{s+1}-\frac{1}{s+3}$.

Cool! $a=\frac{1}{2}$ , $b=2$ , and $a=\frac{3}{2}$.

For fun we could solve $\frac{1}{2}\ddot{v}+2\dot{v}+\frac{3}{2}s=u(t)$

Using Laplace transforms with rest conditions $v(0)=0$ and $\dot{v}(0)=0$

Applying partial fractions

Finally