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:
Write down the characteristic equation
This is a quadratic equation. Let
If
and
and
be its roots we have
;
are distinct real numbers (this happens if
), then the general solution is
(3)
If
If
(which happens if
and
), then the general solution is
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:
Characteristic equation and its roots
Since 4-8 = -4<0, we have complex roots
General solution
. Therefore,
and
;
;
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