Solution
= Solution
The <characteristic roots> of the homogeneous <linear ordinary differential equation> are $2$ and $3$. The forcing is resonant with the $e^{3x}$ solution, so the <method of undetermined coefficients> uses the particular solution $xe^{3x}$. Thus $y=Ae^{2x}+Be^{3x}+xe^{3x}$. The two initial conditions are $A+B=0$ and $2A+3B+1=0$, giving
$$
\boxed{y=e^{2x}+(x-1)e^{3x}.}
$$