4.3 SOLUTION OF DIFFERENTIAL AND INTEGRO-DIFFERENTIAL EQUATIONS

The time-differentiation property of the Laplace transform has set the stage for solving linear differential (or integro-differential) equations with constant coefficients. Because d ^ky/ dt ^k s ^kY( s), the Laplace transform of a differential equation is an algebraic equation that can be readily solved for Y( s). Next we take the inverse Laplace transform of Y( s) to find the desired solution y( t). The following examples demonstrate the Laplace transform procedure for solving linear differential equations with constant coefficients.

EXAMPLE 4.10

Solve the second-order linear differential equation

for the initial conditions y(0 ^?) = 2 and and the input x( t) = e ^{?4 t} u( t).

The equation is

Let

Then from Eqs. (4.24)

and

Moreover, for x( t) = e ^{?4 t} u( t),

Taking the Laplace transform of Eq. (4.35b), we obtain

Collecting all the terms of Y( s) and the remaining terms separately on the left-hand side, we obtain

Therefore

and

Expanding the right-hand side into partial fractions yields

The inverse Laplace transform of this equation yields

Example 4.10 demonstrates the ease with which the Laplace transform can solve linear differential equations with constant coefficients. The method is general and can solve a linear differential equation with constant coefficients of any order.

Zero-Input and Zero-State Components of Response

The Laplace transform method gives...