Question

Solve (D2 + a2)y = sinax

Answer 1

Let us solve the differential equation $(D^2 + a^2)y = \sin ax.$ The characteristic equation $k^2+a^2=0$ has the solutions $k_1=ai, k_2=-ai,$ and hence the general solution of the homogeneous differential equation is $y=C_1\cos ax+ C_2\sin ax.$ The particular solution of non-homogeneous equation is of the form $y_p=x(L\sin ax+ M\cos ax).$ Then $y_p'=L\sin ax+ M\cos ax+x(aL\cos ax-a M\sin ax),$ and $y_p''=aL\cos ax-a M\sin ax+aL\cos ax-a M\sin ax+x(-a^2L\sin ax-a^2 M\cos ax).$

It follows that $2aL\cos ax-2a M\sin ax+x(-a^2L\sin ax-a^2 M\cos ax)+a^2x(L\sin ax+ M\cos ax)=\sin ax,$

and hence $2aL=0$ and $-2aM=1.$ Therefore, if $a\ne 0$ then $L=0,M=-\frac{1}{2a}.$

We conclude that if if $a\ne 0$ then the general solution of the differential equation $(D^2 + a^2)y = \sin ax$ is $y=C_1\cos ax+ C_2\sin ax-\frac{1}{2a}x\cos ax.$ If $a=0$ then the general solution of the differential equation $D^2 y = 0$ is $y=C_1x+C_2.$