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. (2.91) is expanded by means of partial fractions as discussed previously, the following expansion is obtained:
The inverse Laplace transform of Eq. (2.92) is given, using Table 2.1 or Appendix A, by
This solution is composed of two portions: the steady-state solution given by 1, and the transient solution given by −4e3t + 5e−2t. As a check of the steady-state solution, we can apply the final-value theorem given by Eq. (2.74):
Eample 2. As a second example, the following differential equation is considered and the inapplicability of the final-value theorem when the function is not analytic in the right half-plane is illustrated:
The initial conditions are assumed to be
Taking the Laplace transform of both sides of Eq. (2.95), the following equation is obtained:
By substituting the values of the initial conditions, and solving for Y(s), the following equation is obtained:
Expansion of Eq. (2.98) by means of partial fractions, as discussed previously, gives
From Table 2.1 or Appendix A, the inverse Laplace transform of Eq. (2.99) is given by
It is obvious from Eq. (2.100) that the final value of this function is infinite. However, if one were to apply the final-value theorem to Eq. (2.98), the incorrect final value of 1.75 would be obtained. This example, therefore, illustrates very clearly that Eq. (2.74) cannot be applied when the function F(s) is not analytic in the right half-plane.