Question

Given that p(x) = x is a particular solution to the

differential equation y''+ y = x write the generalized sotlution and check by verifying that the solution satisfies

the equation

Answer 1

Let us solve the differential equation $y''+ y = x.$ The characteristic equation $k^2+1=0$ has the roots $k_1=i$ and $k_2=-i.$ Taking into account that $p(x) = x$ is a particular solution, we conclude that the general solution is $y=C_1\cos x+C_2\sin x+x.$

Let us show that the solution satisfies the equation. Since $y'=-C_1\sin x+C_2\cos x+1,\ y''=-C_1\cos x-C_2\sin x,$ and

$y''+ y=-C_1\cos x-C_2\sin x+C_1\cos x+C_2\sin x+x=x,$

we conclude that $y=C_1\cos x+C_2\sin x+x$ is indeed the solution of the equation.