Chapter Contents

• Laplace Transforms
• 1a. The Unit Step Function - Definition
• Oliver Heaviside
• 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
• Laplace Transforms Problem Solver

• Comments, Questions?

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7. The Inverse Laplace Transform

Definition

If G(s) = {g(t)}, then the inverse transform of G(s) is defined as:

^-1G(s) = g(t)

Some Properties of the Inverse Laplace Transform

We first saw these properties in the Table of Laplace Transforms.

Property 1: Linearity Property

^-1{a G₁(s) + b G₂(s)} = a g₁(t) + b g₂(t)

Property 2: Shifting Property

If ^-1G(s) = g(t), then ^-1G(s - a) = e^atg(t)

Property 3

If ^-1G(s) = g(t), then

Property 4

If ^-1G(s) = g(t), then ^-1{e^-asG(s)} = u(t - a) • g(t - a)

EXAMPLES

Find the inverse of the following transforms and sketch the functions so obtained.

(a)

Answer

(b)

Answer

(c)

Answer

(d)

Answer

(e)

Answer

(f)

Answer

(g) (where T is a constant)

Answer

Examples Involving Partial Fractions

We first met Partial Fractions in the Methods of Integration section. You may wish to revise partial fractions before attacking this section.

Obtain the inverse Laplace transforms of the following functions:

(h)

Answer

(i)

Answer

Integral and Periodic Types

(j)

Answer

(k)

Answer

(l)

Answer