= Solution
For the drifted <Brownian motion> $Y_t=y+B_t+at$ on $(0,\infty)$, the <diffusion generator> is $\mathcal L=\frac12\frac{d^2}{dz^2}+a\frac{d}{dz}$. Seek a bounded solution of $\mathcal Lu=\lambda u$ with $u(0)=1$. The exponential ansatz $u(z)=e^{rz}$ gives
$$
\frac12r^2+ar=\lambda,\qquad r=-a\pm\sqrt{a^2+2\lambda}.
$$
Because $\lambda>0$, the plus root is positive and the minus root is negative. Boundedness on $[0,\infty)$ therefore selects
$$
u(z)=\exp\left(-z\left(a+\sqrt{a^2+2\lambda}\right)\right).
$$
It satisfies the boundary condition and all hypotheses of part (d). Hence
$$
\boxed{\mathbb E e^{-\lambda S}=\exp\left(-y\left(a+\sqrt{a^2+2\lambda}\right)\right).}
$$
This is the <first-passage Laplace transform for Brownian motion with drift>, with $e^{-\lambda\infty}=0$. At $a=0$ it reduces to $e^{-y\sqrt{2\lambda}}$. As $\lambda\downarrow0$, it gives $\mathbb P(S<\infty)=e^{-2y\max(a,0)}$, agreeing with certainty of hitting when the drift points towards zero and a possible escape when it points away.
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