6.33 Example continued

To obtain the solution to the inhomogeneous equation, we just have to find the PI. Since $q(x)$ is a quadratic polynomial, then we have to look for a PI of the form

To find the values of $b_{2},b_{1},b_{0}$ we therefore just plug this polynomial into the differential equation to see what we obtain. We have $y^{\prime}={2b_{2}x+b_{1}}$ and $y^{\prime\prime}={2b_{2}}$. To obtain equality in the differential equation, we therefore require:

Then we see immediately that $b_{2}={1}$ and therefore, inspecting the linear term, that $b_{1}={\frac{-5}{3}}$, and hence finally that $b_{0}={\frac{22}{18}=\frac{11}{9}.}$ Adding PI and CF we obtain the general solution: