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

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Systems of differential equations can be solved using powerful linear algebra tools. For linear systems, eigenvalue-eigenvector methods and the matrix exponential provide explicit formulas for the solution. Autonomous systems require setting the derivatives to zero to find critical points, which reveal equilibrium behavior. Mastery of these techniques is essential for analyzing both mechanical and biological models, and understanding the qualitative behavior of nonlinear systems.

Learning Objectives

1

Understand how to solve systems of linear differential equations using eigenvalue–eigenvector methods and matrix exponentials.

2

Learn to compute critical points in autonomous nonlinear systems and analyze phase plane structure.

3

Apply techniques for diagonalization and variation of parameters to obtain general and particular solutions.

4

Develop skills in interpreting and solving exercises involving both homogeneous and nonhomogeneous systems.

Key Concepts

CONCEPT

DEFINITION

System of Differential Equations

A collection of two or more differential equations involving multiple interdependent functions and their derivatives.

Eigenvalue and Eigenvector

In the context of linear systems, an eigenvalue is a scalar that indicates the factor by which an eigenvector is scaled when acted upon by a matrix. Finding these helps in decoupling and solving the system.

Matrix Exponential

A function defined for a square matrix A as e^(At) that provides a compact representation of the general solution to the homogeneous linear system X' = AX.

Autonomous System

A differential equation or system of differential equations in which the independent variable (often time) does not appear explicitly in the function defining the system.

Critical Point

A point (or equilibrium point) at which all derivatives vanish; for an autonomous system X' = F(X), these are values X such that F(X) = 0.

Example Problems

Example 1

Let $\mathbf{X}=\left(\begin{array}{l}x \\ y\end{array}\right) .$ Then $\mathbf{X}^{\prime}=\left(\begin{array}{rr}3 & -5 \\ 4 & 8\end{array}\right) \mathbf{X}$

Example 2

$\operatorname{Let} \mathbf{X}=\left(\begin{array}{l}x \\ y\end{array}\right) .$ Then $\mathbf{X}^{\prime}=\left(\begin{array}{rr}4 & -7 \\ 5 & 0\end{array}\right) \mathbf{X}$

Example 3

$\operatorname{Let} \mathbf{X}=\left(\begin{array}{l}x \\ y \\ z\end{array}\right) .$ Then $\mathbf{X}^{\prime}=\left(\begin{array}{rrr}-3 & 4 & -9 \\ 6 & -1 & 0 \\ 10 & 4 & 3\end{array}\right) \mathbf{X}$

Example 4

$\operatorname{Let} \mathbf{X}=\left(\begin{array}{l}x \\ y \\ z\end{array}\right) .$ Then $\mathbf{X}^{\prime}=\left(\begin{array}{rrr}1 & -1 & 0 \\ 1 & 0 & 2 \\ -1 & 0 & 1\end{array}\right) \mathbf{X}$

Example 5

$\operatorname{Let} \mathbf{X}=\left(\begin{array}{l}x \\ y \\ z\end{array}\right) .$ Then $\mathbf{X}^{\prime}=\left(\begin{array}{rrr}1 & -1 & 1 \\ 2 & 1 & -1 \\ 1 & 1 & 1\end{array}\right) \mathbf{X}+\left(\begin{array}{r}0 \\ -3 t^{2} \\ t^{2}\end{array}\right)+\left(\begin{array}{r}t \\ 0 \\ -t\end{array}\right)+\left(\begin{array}{r}-1 \\ 0 \\ 2\end{array}\right)$
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Step-by-Step Explanations

QUESTION

For the autonomous system x' = y and y' = -9 sin(x), find all the critical points.

STEP-BY-STEP ANSWER:

Step 1: Set the derivatives equal to zero. That is, set x' = y = 0 and y' = -9 sin(x) = 0.
Step 2: From y = 0, we already have one condition on the critical point.
Step 3: Solve -9 sin(x) = 0 which implies sin(x) = 0.
Step 4: Recall that sin(x) = 0 when x = nπ, where n is any integer.
Final Answer: The critical points are (nπ, 0) for n = 0, ±1, ±2, …

Finding Critical Points of an Autonomous System

QUESTION

Given a system X' = AX, explain briefly how eigenvalues and eigenvectors are used to diagonalize A and find the general solution.

STEP-BY-STEP ANSWER:

Step 1: Compute the eigenvalues λ by solving det(A - λI) = 0.
Step 2: For each eigenvalue, find a corresponding eigenvector K.
Step 3: Form the matrix P whose columns are the independent eigenvectors.
Step 4: Diagonalize A as A = P D P^(-1) where D is the diagonal matrix of eigenvalues.
Step 5: The general solution is then obtained via X(t) = P e^(Dt) P^(-1) C, where C is the constant vector determined by initial conditions.
Final Answer:

Solving a Linear System by Diagonalization

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Common Mistakes

  • Failing to set all derivatives equal to zero when finding critical points in autonomous systems.
  • Mixing up signs when computing eigenvalues or during matrix operations, which can lead to incorrect conclusions about system stability.
  • Overlooking the possibility of complex eigenvalues and misinterpreting oscillatory versus exponential behavior.
  • Not checking the linear independence of eigenvectors, potentially resulting in an incomplete set of solutions.