> [!theorem] Strategy > > Let $(L)$ be a [[Second Order ODE|second order ODE]] with only constant coefficients, then > $ > a\frac{d^2y}{dt^2} + b\frac{dy}{dt} + cy = 0 > $ > Suppose that the solution has a form $e^{rt}$, then we have > $ > 0 = (ar^2 + br + c)e^{rt} > $ > which allows solving $r$ as a quadratic equation, whose solutions depend on the [[Discriminant|discriminant]]. > > Which allows obtaining the general solution > $ > y = e^{\alpha t}\paren{k_1 \cos(\beta t) + k_2 \sin (\beta t)} > $ > when the equation has no repeated roots, and > $ > y = (k_1 t + k_2)e^{\alpha t} > $ > when it has repeated roots.