 Solution

For the homogeneous linear ordinary differential equation, the characteristic equation isIts real solution is $y_h=e^{-x}(C_1\cos x+C_2\sin x)$. For a particular solution to the full ordinary differential equation, try $y_p=ax+b$. Substitution givesso $a=4$ and $b=-2$. Adding the homogeneous and particular solutions yieldsThe two arbitrary real constants account for the second order of the ordinary differential equation.