Solving LCCDEs by Unilateral Laplace Transform

Due to its differentiation property, the unilateral Laplace transform is a powerful tool for solving LCCDEs with arbitrary initial conditions.

Example: A system is described by this LCCDE

with initial conditions

To find , we first apply unilateral Laplace transform to the differential equation to get

which can be then solved for algebraically to get

This is the general solution of the LCCDE which is composed of two parts:
• The homogeneous (zero-input) solution caused by the nonzero initial conditions (, ) with zero input (a DE is homogeneous if its right-hand side representing input is zero):

• The particular (zero-state) solution caused by the zero initial conditions () with nonzero input :

Given specific values for , and such as , and , can be partial fraction expanded to be

Finally, the inverse Laplace transform of gives the time domain solution

The particular solution under zero initial condition (caused by the input only) is

which is quite different from the output with non-zero initial conditions.

If bilateral Laplace transform is applied to the same DE, we get

which can be solved for

This is the particular solution above with zero initial conditions. From this we see that bilateral Laplace transform can only solve systems of zero initial conditions. When the initial conditions of the system are not all zero, unilateral Laplace transform has to be used.