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 section covers advanced techniques in analyzing autonomous systems through the identification of critical points, linearization, and use of the Jacobian matrix to determine stability. It also introduces methods for studying periodic solutions, including the use of polar coordinates and separation of variables. Moreover, the concepts of orthogonal functions are highlighted in the context of Fourier series, with applications ranging from mechanical models like pendulums to ecological population models. A thorough understanding of these techniques is essential for modeling real-world phenomena and solving complex differential equations.

Learning Objectives

1

Identify and compute critical points of nonlinear autonomous systems and use them to infer the behavior of solutions.

2

Apply linearization techniques and analyze the Jacobian matrix to classify the stability of equilibrium points.

3

Utilize polar coordinate transformations and separation of variables to study periodic solutions, limit cycles, and spiraling behavior.

4

Interpret autonomous systems as mathematical models (e.g., pendulums, predator–prey dynamics) and connect them with physical phenomena.

5

Understand the concept of orthogonal functions in the context of Fourier series and practice evaluating integrals that demonstrate orthogonality.

Key Concepts

CONCEPT

DEFINITION

Autonomous System

A system of differential equations in which the derivatives depend only on the state variables, not explicitly on time.

Critical (Equilibrium) Point

A point in the phase plane where the derivatives vanish, indicating a state of equilibrium.

Stability

A property describing how solutions behave in the vicinity of a critical point. It includes concepts such as stable, unstable, and saddle points.

Linearization

The process of approximating a nonlinear system near a critical point by its tangent linear system, usually using the Jacobian matrix.

Jacobian Matrix

A matrix of first-order partial derivatives used to approximate the behavior of a nonlinear system near an equilibrium.

Limit Cycle

A closed trajectory in the phase plane that represents a periodic solution, attracting nearby solutions under certain conditions.

Orthogonal Functions

Functions that satisfy an integral zero inner product condition over a given domain; fundamental in developing Fourier series.

Fourier Series

An expansion of a periodic function in terms of sines, cosines, or complex exponentials, which are orthogonal basis functions.

Example Problems

Example 1

The corresponding plane autonomous system is \[ x^{\prime}=y, \quad y^{\prime}=-9 \sin x \] If $(x, y)$ is a critical point, $y=0$ and $-9 \sin x=0 .$ Therefore $x=\pm n \pi$ and so the critical points are $(\pm n \pi, 0)$ for $n=0,1,2, \dots$

Example 2

The corresponding plane autonomous system is \[ x^{\prime}=y, \quad y^{\prime}=-2 x-y^{2} \] If $(x, y)$ is a critical point, then $y=0$ and so $-2 x-y^{2}=-2 x=0 .$ Therefore (0,0) is the sole critical point.

Example 3

The corresponding plane autonomous system is $$ x^{\prime}=y, \quad y^{\prime}=x^{2}-y\left(1-x^{3}\right) $$

Example 4

The corresponding plane autonomous system is \[ x^{\prime}=y, \quad y^{\prime}=-4 \frac{x}{1+x^{2}}-2 y \] If $(x, y)$ is a critical point, $y=0$ and $\operatorname{so}-4 x /\left(1+x^{2}\right)-2(0)=0 .$ Therefore $x=0$ and so (0,0) is the sole critical point.

Example 5

The corresponding plane autonomous system is \[ x^{\prime}=y, \quad y^{\prime}=-x+\epsilon x^{3} \] If $(x, y)$ is a critical point, $y=0$ and $-x+\epsilon x^{3}=0 .$ Hence $x\left(-1+\epsilon x^{2}\right)=0$ and so $x=0, \sqrt{1 / \epsilon},-\sqrt{1 / \epsilon} .$ The critical points are $(0,0),(\sqrt{1 / \epsilon}, 0)$ and $(-\sqrt{1 / \epsilon}, 0)$
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Step-by-Step Explanations

QUESTION

For the autonomous system defined by x' = y and y' = -9 sin x, determine all critical points.

STEP-BY-STEP ANSWER:

Step 1: Recognize that a critical (equilibrium) point occurs when both derivatives are zero.
Step 2: Set x' = y = 0. This immediately gives the condition y = 0.
Step 3: Substitute y = 0 into the second equation to obtain y' = -9 sin x = 0.
Step 4: Solve -9 sin x = 0, which simplifies to sin x = 0.
Step 5: Determine the solutions of sin x = 0: x = nπ, where n is any integer.
Final Answer: All critical points are given by (x, y) = (nπ, 0) for n = 0, ±1, ±2, …

Finding Critical Points in an Autonomous System

QUESTION

Show that for m ≠ n, the sine functions sin((2n+1)x) and sin((2m+1)x) are orthogonal on the interval [0, π/2].

STEP-BY-STEP ANSWER:

Step 1: Write the integral: ∫₀^(π/2) sin((2n+1)x) sin((2m+1)x) dx.
Step 2: Use the product-to-sum formula for sines: sin A sin B = ½[cos(A−B) − cos(A+B)].
Step 3: The integral becomes ½ ∫₀^(π/2) [cos(2(n−m)x) − cos(2(n+m+1)x)] dx.
Step 4: Evaluate each cosine integral; since m ≠ n, both integrals result in sine functions evaluated at the boundaries that vanish.
Final Answer: The integral evaluates to 0, hence the sine functions are orthogonal on [0, π/2].

Verifying Orthogonality of Sine Functions

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Common Mistakes

  • Failing to set both components of the system (e.g., x' and y') to zero when finding critical points.
  • Incorrectly computing or sign errors in the Jacobian matrix during linearization.
  • Mixing up the conditions for stability (stable node, spiral, saddle) due to misinterpretation of the trace and determinant.
  • Overlooking the domain restrictions when applying orthogonality conditions in Fourier series.
  • Confusing periodic solutions with centers; not all centers guarantee periodic trajectories in nonlinear systems.