= Solution
Factor the differential operator as $(D-a)(D-b)$, where $D=d/dx$. Its homogeneous solutions are $e^{ax},e^{bx}$, while a nonresonant particular solution is $e^{cx}/[(c-a)(c-b)]$. Imposing the two zero initial conditions gives the <exponential forcing response with zero initial data>
$$
\boxed{y(x)=\frac{e^{ax}}{(a-b)(a-c)}+\frac{e^{bx}}{(b-a)(b-c)}+\frac{e^{cx}}{(c-a)(c-b)}.}
$$
One direct derivation avoids a large simultaneous system. Set $v=y'-by$, so $v'-av=e^{cx}$ and $v(0)=0$. The <integrating factor> gives $v=(e^{cx}-e^{ax})/(c-a)$; solving $y'-by=v$ with $y(0)=0$ produces the displayed expression. This also verifies the zero derivative at the origin, since $v(0)=y(0)=0$.
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