= Solution
{c}
The deterministic <stochastic integral> in the solution is centered Gaussian with variance
$$
v(t)=\int_0^ta(s)^{-2}\sigma(s)^2ds.
$$
Therefore $X(t,x)$ has mean $a(t)x$ and variance
$$
V(t)=a(t)^2v(t)
=\int_0^t e^{2\int_s^t\mu(r)dr}\sigma(s)^2ds.
$$
For $t>0$ this is strictly positive, since $\sigma>0$. Its density at $y$ is
$$
\boxed{\frac1{\sqrt{2\pi V(t)}}
\exp\left[-\frac{(y-a(t)x)^2}{2V(t)}\right].}
$$
The boundedness assumptions on $\sigma$ and its reciprocal in particular ensure nondegeneracy; compact-interval <continuity> is already sufficient for the displayed finite-time construction.
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