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