Question

Given that p(x) = x is a particular solution to the differential equation y'' + y = x write the generalized solution and check by verifying that the solution satisfies the equation.

Answer 1

$\text{The characteristic equation for the given differential equation is given by}\\ y^2+1 = 0\\ \implies y = \pm i\\ \text{Hence the complimentary solution is given by}\\ y = C_1e^{ix} + C_2e^{-ix}\\ \implies y_c(x) = C_1\sin x + C_2 \cos x\\ \text{The general solution is given by $y_c(x) + y_p(x)$}\\ \text{$y_p(x)$ is already given, that is, x}\\ \implies y = C_1 \sin x + C_2 \cos x+ x\\ \text{Let y(x) = x be a solution, therefore $y'(x)=1$ and $y''(x) = 0$}\\ \text{Hence substituting into the given differential equation, we have}\\ y'' + y = x\\ \text{Showing that y = x is a solution of the given differential equation}$