Book cover for Advanced Engineering Mathematics

Advanced Engineering Mathematics

Dennis G. Zill, Michael R. Cullen

ISBN #9780763740955

3rd Edition

4,310 Questions

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17,647 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This textbook section covers a wide array of topics central to solving differential equations using the Laplace transform and series solutions. Key areas include the definition and inversion of the Laplace transform, translation theorems, handling the Dirac delta function for impulsive forces, and methods to determine convergence of power series solutions. By applying these techniques, one can transform complex differential and integral problems into more accessible algebraic ones. Understanding these concepts is essential for solving linear ODEs, systems of equations, and modeling real-world dynamical systems.

Learning Objectives

1

Apply the definition and properties of the Laplace transform and its inverse to solve differential equations and integral equations.

2

Demonstrate how to use translation (shifting) theorems and operational properties in the Laplace transform domain.

3

Analyze and determine the convergence of power series and series solutions about ordinary points.

4

Solve systems of linear differential equations using Laplace transform methods and series methods.

5

Interpret applications of the Dirac delta function and understand its role in modeling impulsive forces.

Key Concepts

CONCEPT

DEFINITION

Laplace Transform

A transformation defined by F(s) = βˆ«β‚€βˆž e^(βˆ’st) f(t) dt, used to convert differential equations in the time domain into algebraic equations in the s-domain.

Inverse Laplace Transform

A process to recover the original time-domain function f(t) from its Laplace transform F(s), often using partial fractions, convolution, or transform tables.

Translation (Shifting) Theorem

A property that relates the Laplace transform of a shifted function f(t βˆ’ a) multiplied by the unit step to e^(βˆ’as)F(s), which simplifies handling of delayed responses.

Dirac Delta Function

A generalized function Ξ΄(t βˆ’ a) that is zero everywhere except at t = a and satisfies βˆ«β‚‹βˆžβˆž Ξ΄(t βˆ’ a) dt = 1, used to model instantaneous impulses.

Radius of Convergence

A number R such that a power series converges absolutely when |x βˆ’ xβ‚€| < R and diverges for |x βˆ’ xβ‚€| > R.

Series Solution About Ordinary Points

A method for solving differential equations in which the solution is expressed as a power series centered at an ordinary point, with convergence determined by the nearest singularity.

Example Problems

Example 1

$$\begin{aligned} \mathscr{L}\{f(t)\} &=\int_{0}^{1}-e^{-s t} d t+\int_{1}^{\infty} e^{-s t} d t=\left.\frac{1}{s} e^{-s t}\right|_{0} ^{1}-\left.\frac{1}{s} e^{-s t}\right|_{1} ^{\infty} \\ &=\frac{1}{s} e^{-s}-\frac{1}{s}-\left(0-\frac{1}{s} e^{-s}\right)=\frac{2}{s} e^{-s}-\frac{1}{s}, \quad s>0 \end{aligned}$$

Example 2

$$\mathscr{L}\{f(t)\}=\int_{0}^{2} 4 e^{-s t} d t=-\left.\frac{4}{s} e^{-s t}\right|_{0} ^{2}=-\frac{4}{s}\left(e^{-2 s}-1\right), \quad s>0$$

Example 3

$$\begin{aligned} \mathscr{L}\{f(t)\} &=\int_{0}^{1} t e^{-s t} d t+\int_{1}^{\infty} e^{-s t} d t=\left.\left(-\frac{1}{s} t e^{-s t}-\frac{1}{s^{2}} e^{-s t}\right)\right|_{0} ^{1}-\left.\frac{1}{s} e^{-s t}\right|_{1} ^{\infty} \\ &=\left(-\frac{1}{s} e^{-s}-\frac{1}{s^{2}} e^{-s}\right)-\left(0-\frac{1}{s^{2}}\right)-\frac{1}{s}\left(0-e^{-s}\right)=\frac{1}{s^{2}}\left(1-e^{-s}\right), \quad s>0 \end{aligned}$$

Example 4

$$\begin{aligned} \mathscr{L}\{f(t)\} &=\int_{0}^{1}(2 t+1) e^{-s t} d t=\left.\left(-\frac{2}{s} t e^{-s t}-\frac{2}{s^{2}} e^{-s t}-\frac{1}{s} e^{-s t}\right)\right|_{0} ^{1} \\ &=\left(-\frac{2}{s} e^{-s}-\frac{2}{s^{2}} e^{-s}-\frac{1}{s} e^{-s}\right)-\left(0-\frac{2}{s^{2}}-\frac{1}{s}\right)=\frac{1}{s}\left(1-3 e^{-s}\right)+\frac{2}{s^{2}}\left(1-e^{-s}\right), \quad s>0 \end{aligned}$$

Example 5

$$\begin{aligned} \mathscr{L}\{f(t)\} &=\int_{0}^{\pi}(\sin t) e^{-s t} d t=\left.\left(-\frac{s}{s^{2}+1} e^{-s t} \sin t-\frac{1}{s^{2}+1} e^{-s t} \cos t\right)\right|_{0} ^{\pi} \\ &=\left(0+\frac{1}{s^{2}+1} e^{-\pi s}\right)-\left(0-\frac{1}{s^{2}+1}\right)=\frac{1}{s^{2}+1}\left(e^{-\pi s}+1\right), \quad s>0 \end{aligned}$$
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Step-by-Step Explanations

QUESTION

Given a power series βˆ‘ aβ‚™ (x βˆ’ 5)ⁿ where aβ‚™ = 1/10ⁿ, determine the radius of convergence.

STEP-BY-STEP ANSWER:

Step 1: Write the ratio of consecutive terms: |aβ‚™β‚Šβ‚ (x βˆ’ 5)ⁿ⁺¹ / (aβ‚™ (x βˆ’ 5)ⁿ)| = |1/10ⁿ⁺¹ (x βˆ’ 5)ⁿ⁺¹ / (1/10ⁿ (x βˆ’ 5)ⁿ)| = |(x βˆ’ 5)|/10.
Step 2: For convergence, the ratio must be less than 1: |(x βˆ’ 5)|/10 < 1.
Step 3: Multiply both sides by 10: |x βˆ’ 5| < 10.
Final Answer: The radius of convergence is R = 10, and the series converges for x in the interval (βˆ’5, 15) (after checking endpoints if necessary).

Using the Ratio Test to determine the radius of convergence

QUESTION

How do you transform f(t βˆ’ k){t βˆ’ k} if its Laplace transform of f(t) is F(s)?

STEP-BY-STEP ANSWER:

Step 1: Recognize that the translation theorem states: {f(t βˆ’ k) U(t βˆ’ k)} = e^(βˆ’ks) F(s), where U is the unit step function.
Step 2: In some texts the notation {f(t)}(t βˆ’ k) indicates that the power series or transform is taken about the shifted point.
Step 3: Therefore, if f(t) has Laplace transform F(s), then the Laplace transform of f(t βˆ’ k) (times the appropriate unit step function) is e^(βˆ’ks) F(s).
Final Answer: {f(t βˆ’ k) U(t βˆ’ k)} = e^(βˆ’ks) F(s), which is a crucial property for handling delayed responses in differential equations.

Applying the Translation Theorem in Laplace Transforms

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Common Mistakes

  • Failing to include the exponential factor (e^(?as)) when using the translation theorem, which leads to incorrect transforms.
  • Ignoring the conditions of convergence when applying the ratio test to power series, particularly mishandling endpoint analysis.
  • Overlooking the need to check initial conditions post-inversion in Laplace transform problems, especially when dealing with impulsive inputs like the Dirac delta function.
  • Confusing operational properties (like differentiation in the s-domain) with properties of the original function, resulting in algebraic mistakes in the transform process.