Lecture 5.2: Properties and applications of the Laplace transform - - PowerPoint PPT Presentation

Lecture 5.2: Properties and applications of the Laplace transform

Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 2080, Differential Equations

• M. Macauley (Clemson)

Lecture 5.2: Properties & applications of L Differential Equations 1 / 7

Laplace transform fundamentals

Two key properties

L is linear. L turns derivatives into multiplication.

• M. Macauley (Clemson)

Lecture 5.2: Properties & applications of L Differential Equations 2 / 7

Useful shortcuts

More properties

Suppose we know F(s) = L{f (t)}. Then: (i) L{eatf (t)} = F(s − a); (ii) L{tf (t)} = −F ′(s); (iii) L{tnf (t)} = (−1)n · dn

dsn F(s).

Examples

(i) Compute the Laplace transform of f (t) = e2t cos 3t. (ii) Compute the Laplace transform of f (t) = t2e3t.

• M. Macauley (Clemson)

Lecture 5.2: Properties & applications of L Differential Equations 3 / 7

Using the Laplace transform to solve ODEs

Example

Sove the initial value problem y ′′ − y = e2t, y(0) = 0, y ′(0) = 1.

• M. Macauley (Clemson)

Lecture 5.2: Properties & applications of L Differential Equations 4 / 7

Inverse Laplace transforms

Example

Compute L−1

1 s2+4s+13

• .
• M. Macauley (Clemson)

Lecture 5.2: Properties & applications of L Differential Equations 5 / 7

Comparison of old vs. new methods

• M. Macauley (Clemson)

Lecture 5.2: Properties & applications of L Differential Equations 6 / 7

Structure of the solution to an ODE

A generic example

Consider an initial value problem ay ′′ + by ′ + cy = f (t), y(0) = x0, y ′(0) = v0.

• M. Macauley (Clemson)

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