= Solution
Now $c=b$, so the forcing coincides with a homogeneous exponential and requires a <resonant exponential particular solution>. With $v=y'-by$ as above, $v'-av=e^{bx}$ and $v(0)=0$ give $v=(e^{ax}-e^{bx})/(a-b)$. Therefore
$$
(e^{-bx}y)'=\frac{e^{(a-b)x}-1}{a-b}.
$$
Integrating from zero yields
$$
\boxed{y(x)=\frac{e^{ax}-e^{bx}-(a-b)xe^{bx}}{(a-b)^2},\qquad a\ne b.}
$$
Its numerator starts at order $x^2$, ensuring both initial conditions. It is also the finite coalescing-rate limit $c\to b$ of the nonresonant <exponential forcing response with zero initial data>.
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