= Solution
Inserting the <Dirac delta> forcing into the integral gives the <impulse response>
$$
\boxed{y(t)=\frac1\alpha e^{-t/\alpha}H(t).}
$$
It is zero before the impulse and obeys the homogeneous equation afterward. Integrating the equation across zero gives $\alpha[y(0^+)-y(0^-)]=1$, exactly the jump $1/\alpha$ in this formula. Thus the formula also verifies the <differential equation> as a <distribution>; assigning a particular value at the jump does not affect that distribution.
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