Linear Differential Equations with Constant Coefficients

Abstract

We start with the simple remark that, if x(t) = e ^αt, then x′(t) = αe ^αt, x″(t) = α² e ^αt, hence x″(t) = α²x(t); in the same way, if x(t) = e ^-αt, then x′(t) = -α e ^-αt, x″(t) = α² e ^-αt and again x″(t) = α² x(t). We deduce that the functions t ↦ e ^αt and t ↦ e ^-αt satisfy the equation x″ = α² x; it can be immediately seen thai, for any real numbers C ₁ and C ₂, the function x defined by x(t) = C ₁ e ^αt+C ₂ e ^-αt satisfies the same differential equation.