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Chapter Contents

• Laplace Transforms
• 1a. The Unit Step Function - Definition
• 1b. The Unit Step Function - Products
• 2. Laplace Transform Definition
• Table of Laplace Transformations
• 3. Properties of Laplace Transform
• 4. Transform of Unit Step Functions
• 5. Transform of Periodic Functions
• 6. Transforms of Integrals
• 7. Inverse of the Laplace Transform
• 8. Using Inverse Laplace to Solve DEs
• 9. Integro-Differential Equations and Systems of DEs
• 10. Applications of Laplace Transform

Following are the original SNB files (.tex or .rap) used in making this chapter. For more information, go to SNB info.

SNB files

1a. The Unit Step Function - Definition (SNB)

1b. The Unit Step Function - Products (SNB)

2. Laplace Transform Definition (SNB)

3. Properties of Laplace Transform (SNB)

4. Transform of Unit Step Functions (SNB)

5. Transform of Periodic Functions (SNB)

6. Transforms of Integrals (SNB)

7. Inverse of the Laplace Transform (SNB)

8. Using Inverse Laplace to Solve DEs (SNB)

9. Integro-Differential Equations and Systems of DEs (SNB)

10. Applications of Laplace Transform (SNB)

• Comments, Questions?

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From the math blog...

9. Solving Integro-Differential Equations

An "integro-differential equation" is an equation that involves both integrals and derivatives of an unknown function.

Using the Laplace transform of integrals and derivatives, an integro-differential equation can be solved.

Similarly, it is easier with the Laplace transform method to solve simultaneous differential equations by transforming both equations and then solve the two equations in the s-domain and finally obtain the inverse to get the solution in the t-domain.

EXAMPLE 1 (Integro-Differential Equation)

Solve the equation for the response i(t), given that

and i(0) = 0.

Answer

EXAMPLE 2 (SIMULTANEOUS DEs)

Solve for x(t) and y(t), given that x(0) = 4, y(0) = 3, and:

Answer

The rectangular plot of the solution is an interesting curve...

While here is a polar plot: