Constant coefficient, linear second order, homogeneous equations take the form
This is a constant coefficient equation becauseand
are all constants.
It is linear because we can writeas
where
is a linear operator because
The equation is second order because of the expressionand 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 formso that
and
Substitution into the equation above gives
We can divide by the non zero factorto give
This equation may have distinct real solutionsand
so that
The equation may have one real solutionIn this case
The equation may have complex solutionsand
In this case
Each solution has arbitrary constantsand
These constants may be found given suitable boundary conditions (if the derivatives are with respect to
then the boundary conditions are called initial conditions).