= Solution
The <characteristic equation> of the associated homogeneous <linear ordinary differential equation> is $m^2+5m+6=(m+2)(m+3)=0$, so the <complementary function> is $C_1e^{-2x}+C_2e^{-3x}$. For a <particular solution>, try $y_p=ax+b$. Substitution gives $6ax+5a+6b=6x$, so $a=1$, $b=-5/6$. The unrestricted solution is therefore
$$
y=C_1e^{-2x}+C_2e^{-3x}+x-\frac56.
$$
Its <derivative> at zero is $-2C_1-3C_2+1$. The single prescribed condition imposes $2C_1+3C_2=1$ and leaves one arbitrary constant. Thus the most general permitted solution is
$$
\boxed{y=Ce^{-2x}+\frac{1-2C}{3}e^{-3x}+x-\frac56,\qquad C\in\mathbb R.}
$$
There is no condition fixing $y(0)$, so the remaining arbitrary constant is essential.
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