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.