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

This section of the textbook introduces key definitions and terminology in differential equations, emphasizing the concepts of order, linearity versus nonlinearity, and the significance of domains of definition. It covers methods for verifying solutions, solving initial-value problems by determining arbitrary constants, and applying existence and uniqueness theorems. Practical applications, ranging from population dynamics to physical systems, demonstrate the broad utility of differential equations in modeling real-world phenomena.

Learning Objectives

1

Identify and classify differential equations by order and linearity, and determine their domains of definition.

2

Apply techniques to verify solutions of differential equations using differentiation and substitution.

3

Solve initial?value problems by determining constants from given conditions.

4

Understand and apply existence, uniqueness, and continuity conditions for differential equation models.

5

Model real-world phenomena using differential equations and interpret the resulting mathematical relationships.

Key Concepts

CONCEPT

DEFINITION

Differential Equation

An equation that relates a function with its derivatives; it characterizes how a quantity changes.

Order

The highest derivative present in the differential equation; for example, a second order equation involves the second derivative.

Linearity

A differential equation is linear if the unknown function and its derivatives appear to the first power and are not multiplied together, otherwise it is nonlinear.

Initial-Value Problem

A problem where the value of the unknown function (and possibly some of its derivatives) is specified at a single point.

Existence and Uniqueness

Theorems that guarantee a unique solution exists for a differential equation under appropriate conditions such as continuity of f(x, y) and its partial derivative with respect to y.

Constant Solution

A solution of a differential equation that is independent of the independent variable; found by letting all derivatives equal zero.

Example Problems

Example 1

Second order; linear.

Example 2

Third order; nonlinear because of $(d y / d x)^{4}$ .

Example 3

Fourth order; linear.

Example 4

Second order; nonlinear because of cos $(r+u)$ .

Example 5

Second order; nonlinear because of $(d y / d x)^{2}$ or $\sqrt{1+(d y / d x)^{2}}$.

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Step-by-Step Explanations

QUESTION

Verify that y = e^(-x/2) is a solution of the differential equation 2y' + y = 0.

STEP-BY-STEP ANSWER:

Step 1: Differentiate y = e^(-x/2) with respect to x using the chain rule. This yields y' = -1/2 e^(-x/2).
Step 2: Multiply y' by 2 to get 2y' = 2(-1/2 e^(-x/2)) = -e^(-x/2).
Step 3: Substitute y and 2y' into the differential equation 2y' + y. This gives -e^(-x/2) + e^(-x/2).
Step 4: Simplify the expression to obtain 0.
Final Answer: Since 2y' + y simplifies to 0, y = e^(-x/2) is indeed a solution of the differential equation.

Verifying a Given Solution

QUESTION

For the differential equation x = c1 cos(t) + c2 sin(t) with initial conditions, determine c1 and c2 if x(0) = -1 and x'(0) = 8.

STEP-BY-STEP ANSWER:

Step 1: Evaluate x(0) = c1 cos(0) + c2 sin(0). Since cos(0)=1 and sin(0)=0, this simplifies to c1 = -1.
Step 2: Differentiate x(t) to get x'(t) = -c1 sin(t) + c2 cos(t).
Step 3: Evaluate x'(0): x'(0) = -c1 sin(0) + c2 cos(0) = c2, because sin(0)=0 and cos(0)=1.
Step 4: Set x'(0) equal to 8 so that c2 = 8.
Final Answer: The solution with these initial conditions is x(t) = -cos(t) + 8 sin(t).

Solving an Initial-Value Problem

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Common Mistakes

  • Confusing the order of a differential equation (e.g., mistaking a second order for a first order equation).
  • Overlooking nonlinearity due to hidden nonlinear terms when rewriting the differential equation in different forms.
  • Incorrectly applying initial conditions, leading to errors in determining arbitrary constants.
  • Ignoring domain restrictions, which can result in solutions that are not valid over the entire interval of interest.
  • Mixing up the roles of independent and dependent variables when verifying if a function is a solution.