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 48 = 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.]
[Geometry]
[Algebra]
[Trigonometry ]
[Calculus]
[Complex Variables]
[Matrix Algebra]
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Author: Mohamed
Amine Khamsi
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