Characteristic Equation

Method

For a second order constant coefficient ODE we can guess a solution of the form

$y(t)=e^{\lambda t}$

By plugging this into our ODE we now will have an equation in terms of $\lambda$ that gives us three distinct cases for the form of the solution to the ODE

Distinct Roots

In this case where the characteristic equation has two distinct roots the solution to the ODE will be

$y(t)=C_1{e}^{{\lambda }_{1}t}+C_2{e}^{{\lambda }_{2}t}$

Double Root

In this case where the characteristic equation has one root with multiplicity two the solution to the ODE will be

$y(t)=C_1{e}^{\lambda t}+C_{2}t{e}^{\lambda t}$

Complex Conjugate Roots

In this case where the characteristic equation has two roots that are complex conjugates of each other ${\lambda }_{\pm }=a\pm ib$ the solution to the ODE will be

$y(t)=C_1{e}^{at}\cos (bt)+C_2{e}^{at}\sin (bt)$