Modern Control System Theory and Design, 2nd Edition

2.10.  LAPLACE-TRANSFORM SOLUTION OF DIFFERENTIAL EQUATIONS

After the physical relationships of a linear system have been described by means of its integrodifferential equation, the analysis of the system’s dynamic behavior can be carried out by solving the equations and incorporating the initial conditions into the solution. Two examples are given in this section to illustrate the application of the Laplace transform to solve a linear differential equation. In general, we take the Laplace transform of each term in the differential equation. This step eliminates time and all of the time derivatives from the original equation and results in an algebraic equation in s. The resulting equation is then solved for the transform of the desired time function. The final step involves obtaining the inverse Laplace transform, which yields the solution directly.

Example 1.  Consider the following linear differential equation:

Assume the initial conditions are

By taking the Laplace transform of both sides of Eq. (2.89), the following equation is obtained [using Eqs. (2.66) and (2.67)]:

Substituting the values of the initial conditions and solving for Y(s) yields the following equation:

If Eq. ...