Constant coefficient, linear second order, homogeneous equations take the form
This is a constant coefficient equation because
and
are all constants.
It is linear because we can write
as
where
is a linear operator because
The equation is second order because of the expression
and homogeneous because there is no function of
on the right hand side.
We can solve equations of this form by assuming a solution of the form
so that
and
Substitution into the equation above gives
We can divide by the non zero factor
to give
This equation may have distinct real solutions
and
so that
The equation may have one real solution
In this case
The equation may have complex solutions
and
In this case
Each solution has arbitrary constants
and
These constants may be found given suitable boundary conditions (if the derivatives are with respect to
then the boundary conditions are called initial conditions).