Solution
= Solution
For the <Heaviside step function> forcing, the integral begins at zero when $t>0$:
$$
y(t)=\frac1\alpha\int_0^t e^{-(t-s)/\alpha}\,ds=1-e^{-t/\alpha}.
$$
Before zero it vanishes. Therefore
$$
\boxed{y(t)=(1-e^{-t/\alpha})H(t).}
$$
This <causal response of a first-order relaxation equation> is continuous at zero and approaches the new constant equilibrium exponentially.