First find the characteristic solution. The characteristic equation is
\(r^2-2r+1=(r-1)^2=0\)
which as one root at \(r=1\) of multiplicity 2. This means the characteristic solution for this ODE is
\(y_c=C_1e^x+C_2xe^x\)
For the nonhomogeneous part, you can try a particular solution of the form
\(y_p=(a_2x^2+a_1x+a_0)e^x+be^{-x}\)
which has derivatives
\({y_p}'=(a_2x^2+(2a_2+a_1)x+a_1+a_0)e^x-be^{-x}\)
\({y_p}''=(a_2x^2+(4a_2+a_1)x+2a_2+2a_1+a_0)e^x+be^{-x}\)
Substituting into the ODE, the left hand side reduces significantly to
\(2a_2e^x+4be^{-x}=2e^x-3e^{-x}\)
and it follows that
\(\begin{cases}2a_2=2\\4b=-3\end{cases}\implies a_2=1,b=-\dfrac34\)
Therefore the particular solution is
\(y_p=x^2e^x-\dfrac34e^{-x}\)
and so the general solution is the sum of the characteristic and particular solutions,
\(y=y_c+y_p\)
\(y=C_1e^x+C_2xe^x+x^2e^x-\dfrac34e^{-x}\)
