Question

Annihilator method obtain a particular solution y"'+y=cosx+sinx on R

Answer 1

Let us write the equation $y'''+y=\cos x+\sin x$ in the operator form: $(D^3+1)y=\cos x+\sin x.$

The annihilator operator for the right part is $D^2+1.$ Then we get $(D^2+1)(D^3+1)y=(D^2+1)(\cos x+\sin x)=-\cos x-\sin x+\cos x+\sin x=0.$

In the annihilator method for finding a particular solution the method of undetermined coefficients is used. Since $i$ and $-i$ are not roots of the characteristic equation $k^3+1=0$ of the homogeneous differential equation $y'''+y=0,$ we conclude that the particular solution of the differentail equation $y'''+y=\cos x+\sin x$ is of the form $y_p=A\cos x+B\sin x.$ Then $y_p'=-A\sin x+B\cos x,\ y_p''=-A\cos x-B\sin x,\ y_p'''=A\sin x-B\cos x.$ Thus we get the equation $A\sin x-B\cos x+A\cos x+B\sin x=\cos x+\sin x,$ which is equivalet to $(A-B)\cos x+(A+B)\sin x=\cos x+\sin x.$ Therefore, $A-B=1$ and $A+B=1,$ and hence $A=1$ and $B=0.$

We conclude that the particular solution is $y_p=\cos x.$