Chapter 6.1, Problem 23E

Let $L\left[y\right]:=y^{‴}+y^{\prime }+xy$, $y_1\left(x\right):=\sin x$, and $y_2\left(x\right):=x$. Verify that $L\left[y_1\right]\left(x\right)=x\sin x$ and $L\left[y_2\right]\left(x\right)=x^2+1$. Then use the superposition principle (linearity) to find a solution to the differential equation:

a. $L\left[y\right]=2x\sin x-x^2-1$.

b. $L\left[y\right]=4x^2+4-6x\sin x$.