Linear constant coefficient differential/difference equations (LCCDE)

 (1)

with initial conditions , we assume the solution takes the form , i.e., , and convert the differential equation above into an algebraic equation:

 (2)

where is the characteristic polynomial. Solving the equation we can find roots in general, and the solution of the differential equation can be found as a linear combination:

 (3)

By letting , we can find the coefficients based on the initial conditions.

To solve an Nth order LCCDE

 (4)

with initial conditions , we assume the solution takes the form , i.e., , and convert the difference equation above into an algebraic equation:

 (5)

where is the characteristic polynomial. Solving the equation we can find roots in general, and the solution of the difference equation can be found as a linear combination:

 (6)

By letting , we can find the coefficients based on the initial conditions.