Apply the method of undetermined coefficients to find a particular solution to the following system.wing system.y" - y' + y = sin x. y" - 3y' + 2y = e^x sin x. y" + y = x cos(2x).

[tex]y''-y'+y=\sin x[/tex]The corresponding homogeneous ODE has characteristic equation [tex]r^2-r+1=0[/tex] with roots at [tex]r=\dfrac{1\pm\sqrt3}2[/tex], thus admitting the characteristic solution[tex]y_c=C_1e^x\cos\dfrac{\sqrt3}2x+C_2e^x\sin\dfrac{\sqrt3}2x[/tex]For the particular solution, assume one of the form[tex]y_p=a\sin x+b\cos x[/tex][tex]{y_p}'=a\cos x-b\sin x[/tex][tex]{y_p}''=-a\sin x-b\cos x[/tex]Substituting into the ODE gives[tex](-a\sin x-b\cos x)-(a\cos x-b\sin x)+(a\sin x+b\cos x)=\sin x[/tex][tex]-b\cos x+a\sin x=\sin x[/tex][tex]\implies a=1,b=0[/tex]Then the general solution to this ODE is[tex]\boxed{y(x)=C_1e^x\cos\dfrac{\sqrt3}2x+C_2e^x\sin\dfrac{\sqrt3}2x+\sin x}[/tex][tex]y''-3y'+2y=e^x\sin x[/tex][tex]\implies r^2-3r+2=(r-1)(r-2)=0\implies r=1,r=2[/tex][tex]\implies y_c=C_1e^x+C_2e^{2x}[/tex]Assume a solution of the form[tex]y_p=e^x(a\sin x+b\cos x)[/tex][tex]{y_p}'=e^x((a+b)\cos x+(a-b)\sin x)[/tex][tex]{y_p}''=2e^x(a\cos x-b\sin x)[/tex]Substituting into the ODE gives[tex]2e^x(a\cos x-b\sin x)-3e^x((a+b)\cos x+(a-b)\sin x)+2e^x(a\sin x+b\cos x)=e^x\sin x[/tex][tex]-e^x((a+b)\cos x+(a-b)\sin x)=e^x\sin x[/tex][tex]\implies\begin{cases}-a-b=0\\-a+b=1\end{cases}\implies a=-\dfrac12,b=\dfrac12[/tex]so the solution is[tex]\boxed{y(x)=C_1e^x+C_2e^{2x}-\dfrac{e^x}2(\sin x-\cos x)}[/tex][tex]y''+y=x\cos(2x)[/tex][tex]r^2+1=0\implies r=\pm i[/tex][tex]\implies y_c=C_1\cos x+C_2\sin x[/tex]Assume a solution of the form[tex]y_p=(ax+b)\cos(2x)+(cx+d)\sin(2x)[/tex][tex]{y_p}''=-4(ax+b-c)\cos(2x)-4(cx+a+d)\sin(2x)[/tex]Substituting into the ODE gives[tex](-4(ax+b-c)\cos(2x)-4(cx+a+d)\sin(2x))+((ax+b)\cos(2x)+(cx+d)\sin(2x))=x\cos(2x)[/tex][tex]-(3ax+3b-4c)\cos(2x)-(3cx+3d+4a)\sin(2x)=x\cos(2x)[/tex][tex]\implies\begin{cases}-3a=1\\-3b+4c=0\\-3c=0\\-4a-3d=0\end{cases}\implies a=-\dfrac13,b=c=0,d=\dfrac49[/tex]so the solution is[tex]\boxed{y(x)=C_1\cos x+C_2\sin x-\dfrac13x\cos(2x)+\dfrac49\sin(2x)}[/tex]