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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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