# Homogeneous Linear Equations with Constant Coefficients

A second order homogeneous equation with constant coefficients is written as

where a, b and c are constant. This type of equation is very useful in many applied problems (physics, electrical engineering, etc..). Let us summarize the steps to follow in order to find the general solution:

(1)
Write down the characteristic equation

This is a quadratic equation. Let and be its roots we have ;

(2)
If and are distinct real numbers (this happens if ), then the general solution is

(3)
If (which happens if ), then the general solution is

(4)
If and are complex numbers (which happens if ), then the general solution is

where

,

that is,

Example: Find the solution to the IVP

Solution: Let us follow the steps:

1
Characteristic equation and its roots

Since 4-8 = -4<0, we have complex roots . Therefore, and ;

2
General solution

;

3
In order to find the particular solution we use the initial conditions to determine and . First, we have

.

Since , we get

From these two equations we get

,

which implies

[Differential Equations] [First Order D.E.] [Second Order D.E.]
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