Solve the differential equation by variation of parameters.

$y^{\text{'}\text{'}}3y^{\text{'}}2y=\frac{1}{2e^x}$

Step $1$

We are given a nonhiomogeneous second$-$order differential equation. Similar to the method of saliting by andetermined coefficients, we frist find the complementary function $y_c$ for the associated homogeneous equation. This time, the particular solution ${}_p$ is based on Wronskian determinanes and the general scilation is $={}_c{}_g$

First, we must find the roots of the auxiliary equation for $y^{\text{'}\text{'}}3y^{\text{'}}2y=0.$

$m^{2}3m2=0$

Solving for $m,$ the roots of the auriliary equation are as follows.

smaller value $m_1=$

larger value $,m_2=$