= Solution
The <characteristic polynomial> of the homogeneous equation is $r^2+7r+12=(r+3)(r+4)$. Thus the <complementary function> is $C_1e^{-3x}+C_2e^{-4x}$. The forcing resonates with the first term, so a <resonant exponential particular solution> is $y_p=Kxe^{-3x}$. Substitution gives $y_p''+7y_p'+12y_p=Ke^{-3x}$, hence $K=2$.
The full solution is $y=(C_1+2x)e^{-3x}+C_2e^{-4x}$. At zero, the two <initial conditions> give $C_1+C_2=1$ and $2-3C_1-4C_2=0$. Solving yields $C_1=2$, $C_2=-1$, so
$$
\boxed{y(x)=(2+2x)e^{-3x}-e^{-4x}.}
$$
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