 Solution

Use the method of undetermined coefficients. Substitution of $y_p=(a+bx)\sin x+(c+dx)\cos x$ givesEquating the coefficients of $x\sin x$, $x\cos x$, $\sin x$, and $\cos x$ to the forcing gives $d=1$, $b=0$, $c=1$, $a=-1$. ThusThe characteristic equation of the homogeneous equation is $(r-1{)}^2=0$, with a repeated root. Its complementary function is $(C_1+C_{2}x)e^x$. Since $y_p(0)=1$ and $y_p^{\prime }(0)=0$, the initial conditions give $C_1=-1$ and $C_2=1$. ThereforeDirect differentiation verifies both zero initial data and the original forcing.