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8. Using Inverse Laplace Transforms to Solve Differential Equations

Laplace Transform of Derivatives

We use the following notation:

(a) If we have the function g(t), then G(s) = G = {g(t)}.

(b) g(0) is the value of the function g(t) at t = 0.

(c) g'(0), g''(0),... are the values of the derivatives of the function at t = 0.

If g(t) is continuous and g'(0), g''(0),... are finite, then

(1)

(2) {g''(t)} = s²G − s g(0) − g'(0)

We saw many of these expressions in the Table of Laplace Transforms.

NOTE: If instead of g(t) we have a function y of x, then Equation (2) would simply become:

{y''(x)} = s²Y − s y (0) − y'(0)

Likewise, if we have an expression for current i and it is a function of t, then the equation would become:

{i''(t)} = s²I − s i(0) − i'(0)

(3) For the n-th derivative,

NOTE: If we have y and it is a function of t, then the notation would become:

Subsidiary Equation

The subsidiary equation is the equation in terms of s, G and the coefficients g'(0), g''(0),... etc ,obtained by taking the transforms of all the terms in a linear differential equation.

The subsidiary equation is expressed in the form G = G(s).

EXAMPLES

Write down the subsidiary equations for the following differential equations and hence solve them.

(a) , given that y = 0 when t = 0.

Answer

Scientific Notebook solution:

This is the way we could go about this problem using Scientific Notebook. You don't need any special software to see it (it is in HTML form.)

Answer

---

(b) Solve , given that

y = 1, , when t = 0.

Answer

Scientific Notebook solution:

Answer

---

(c) , given that

y = -2, , when t = 0.

Answer

Scientific Notebook solution:

Answer

APPLICATION

The current i(t) in an electrical circuit is given by the DE

and i(0) = 0, i'(0) = 0.

Determine the current as a function of t.

Answer

Scientific Notebook solution:

Answer

 

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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

1b. The Unit Step Function - Products

2. Laplace Transform Definition

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

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