Homogeneous Linear Equations with Constant Coefficients
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:
This is a
quadratic equation. Let 
and 
be its roots
we have 
;
and are distinct real numbers (this happens
if 
), then the general solution is
(which happens if 
), then the
general solution is
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:
Since 4-8 = -4<0, we have complex roots 
. Therefore,
and 
;
;
and 
. First, we have
.
Since 
, we get
From these two equations we get
,
which implies
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Author: Mohamed Amine Khamsi