> [!NOTE] Theorem (non-homogeneous second order linear equations with constant coefficients) > Let $a \frac{d^{2}}{dt^{2}}x(t) +b \frac{d}{dt} x(t) + cx(t) = s(t) \tag{1}$be a [[Second Order Linear Scalar Ordinary Differential Equation with Real Coefficients|second order linear ordinary differential equation with real coefficients]]. > >Assume that $x_{p}(t)$ is a solution of $(1).$ Then all [[Solution to Scalar Ordinary Differential Equation|solutions]] to $(1)$ are of the form $x(t)=x_{p}(t)+x_{c}(t)$where $x_{c}(t)$ is the [[Solution to homogenous 2nd order linear scalar ODE with real coefficients|general solution to the homogenous equation]]: $a \frac{d^{2}}{dt^{2}}x(t) +b \frac{d}{dt} x(t) + cx(t) = 0.$ **Proof**: ...