University Maths Notes: Calculus – Solving Second Order Constant Coefficient Linear Differential Equations
The general second order differential equation with constant coefficients takes the form
We solve this in two parts.
First we solve the associated homogeneous equation
using
the substitution
obtaining
the characteristic equation
We
can cancel the common factor
to
obtain
The
roots of this equation may be real or complex.
If they are real and distinct,
and
then
the solution to the homogeneous equation is called the complementary
solution
where
and
are
arbitrary constants.
If there is only one real root
then
If the roots are complex,
and
then
The following general solution can usually be quoted, where again A and B are arbitrary constants.
Next we look for a solution
of
the full equation by choosing a function which 'looks' like
If
choose
If
then
choose
If
then
choose
etc.
This function is substituted into the original equation
to be solved for
and
the coefficients
and
or
found.
Then the general solution
is
found by adding
and
Finally
we find the coefficients
that
appear in the complementary solution
These
are found by substituting boundary conditions into
These
boundary condition consist of simultaneous values of
and
If the equation to be solved is in terms ofand
or
andthese
boundary conditions are called initial conditions.