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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3. Some Properties of Laplace Transforms

We saw some of the following properties in the Table of Laplace Transforms.

Property 1. Constant Multiple

If a is a constant and f(t) is a function of t, then

{a f(t)} = a{f(t)}

Example

{7 sin t} = 7{sin t}

[This is not surprising, since the Laplace Transform is an integral and the same property applies for integrals.]

Property 2. Linearity Property

If a and b are constants while f(t) and g(t) are functions of t, then

{a f(t) + b g(t)} = a{f(t)} + b{g(t)}

Example

{3t + 6t² } = 3 {t} + 6{t²}

Property 3. Change of Scale Property

If {f(t)} = F(s) then

Example

Property 4. Shifting Property (Shift Theorem)

{e^atf(t)} = F(s − a)

Example

{e^3tf(t)} = F(s − 3)

Property 5.

EXAMPLES

Obtain the Laplace transforms of the following functions, using the Table of Laplace Transforms and the properties given above.

(We can, of course, use Scientific Notebook to find each of these. Sometimes it needs some more steps to get it in the same form as the Table).

(a) f(t) = 4t²

Answer

(b) v(t) = 5 sin 4t

Answer

(c) g(t) = t cos 7t

Answer

DEMONSTRATION of PROPERTY 5:

{t f(t)}

For example (c), we could have also used Property 5:

with f(t) = cos 7t.

Now

So

So

This is the same result that we obtained using the formula.

For a reminder on derivatives of a fraction, see Derivatives of Products and Quotients.

---

(d) f(t) = e^2t sin 3t

Answer

DEMONSTRATION OF No 4: SHIFTING PROPERTY

For example (d) we could have used:

{e^atg(t)} = G(s − a)

Let g(t) = sin 3t

So

This is the same result we obtained before for example (d).

---

(e) f(t) = t⁴e^-jt

Answer

(f) f(t) = te^-t cos 4t

Answer

(g) f(t) = t² sin 5t

Answer

(h) f(t) = t³ cos t = t²(t cos t)

Answer

(i) f(t) = cos²3t, given that

Answer