Application of Laplace Transform Integrodifferential Equation

Laplace Transform Integrodifferential Equation

Laplace Transform Integrodifferential Equation is the method of using both integral and derivative for unknown equations using Laplace transform.

Laplace Transform Integrodifferential Equation

The Laplace transform is useful in solving linear integrodifferential equations. Using the differentiation and integration properties of Laplace transforms, each term in the integrodifferential equation is transformed.

Initial conditions are automatically taken into account. We solve the resulting algebraic equation in the s-domain. We then convert the solution back to the time domain by using the inverse transform. The following examples illustrate the process.

Laplace Transform Integrodifferential Equation Examples

1. Use the Laplace transform to solve the differential equation

Laplace Transform Integrodifferential Equation

subject to v(0) = 1, v(0) = −2.

Solution:
We take the Laplace transform of each term in the given differential equation and obtain

Laplace Transform Integrodifferential Equation

Substituting v(0) = 1, v(0) = −2,

Laplace Transform Integrodifferential Equation

or

Laplace Transform Integrodifferential Equation

Hence,

Laplace Transform Integrodifferential Equation

where

Laplace Transform Integrodifferential Equation

Hence,

Laplace Transform Integrodifferential Equation

By the inverse Laplace transform,

Laplace Transform Integrodifferential Equation

 

2. Solve for the response y(t) in the following integrodifferential equation.

Laplace Transform Integrodifferential EquationSolution:
Taking the Laplace transform of each term, we get

Laplace Transform Integrodifferential EquationSubstituting y(0) = 2 and multiplying through by s,

Laplace Transform Integrodifferential Equation

or

Laplace Transform Integrodifferential Equation

where

Laplace Transform Integrodifferential Equation

Thus

Laplace Transform Integrodifferential Equation

Its Laplace inverse transform is

Laplace Transform Integrodifferential Equation