Book cover for Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart

ISBN #9781285741550

8th Edition

6,422 Questions

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2,819,387 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section explains how differential equations are used to model various physical phenomena. It begins with population growth models, introducing both the exponential and logistic models, and highlights the importance of equilibrium and initial conditions. The discussion then extends to mechanical systems, such as the motion of a mass-spring system modeled by a second-order differential equation. Finally, it introduces graphical and numerical methods for studying differential equations when explicit solutions are not available, emphasizing that understanding the qualitative behavior of solutions can be just as important as finding exact formulas.

Learning Objectives

1

Explain how differential equations serve as mathematical models for real-world phenomena such as population growth and mechanical vibrations.

2

Derive and solve first-order differential equations, including the exponential growth model and the logistic differential equation.

3

Analyze equilibrium solutions and understand the implications of initial conditions in selecting particular solutions.

4

Recognize second-order differential equations in physical contexts, such as modeling the motion of a spring, and identify their trigonometric solution forms.

5

Introduce graphical (direction fields) and numerical (Euler’s method) approaches to understanding differential equations when explicit solutions are not available.

Key Concepts

CONCEPT

DEFINITION

Differential Equation

An equation that involves an unknown function and its derivatives; it represents how changes in the system variables relate to the state of the system.

Exponential Growth Model

A first-order differential equation of the form dP/dt = kP with solution P(t) = Ce^(kt), modeling situations where growth rate is proportional to the current value.

Logistic Differential Equation

A model expressed as dP/dt = kP(1 – P/M) that accounts for limited resources by incorporating a carrying capacity M, representing a more realistic population growth.

Equilibrium Solution

A constant solution of a differential equation for which the derivative is zero (dP/dt = 0), indicating no change in the state of the system.

Second-Order Differential Equation

An equation that involves the second derivative of the unknown function, such as m d²x/dt² + 2kx = 0, commonly seen in mechanical vibrations and oscillatory systems.

Initial Value Problem

A differential equation coupled with a specification of the state of the system at a given initial time, used to determine a unique solution from a family of solutions.

Direction Field

A graphical tool that represents the slope of the solution curve at each point in the plane, aiding in visualizing the behavior of differential equations when explicit solutions are complex or unavailable.

Euler’s Method

A numerical technique for approximating solutions of differential equations by stepping forward using the derivative information.

Example Problems

Example 1

Show that $ y = \frac{2}{3}e^x + e^{-2x} $ is a solution of the differential equation $ y^x + 2y = 2e^x $.

Example 2

Verify that $ y = -t \cos t - t $ is a solution of the initial value problem. $ t \frac{ty}{dt} = y + t^{2} \sin t $ $ y = \pi = 0 $

Example 3

(a) For what values of $ r $ does the function $ y = e^{rx} $ satisfy the differential equation $ 2y^{"} + y^{'} - y = 0? $ (b) if $ r_1 $ and $ r_2 $ are the values of $ r $ that you found in part (a), show that every member of the family of functions $ y = ae^{r_1{x}} + be^{r_2{x}} $ is also a solution.

Example 4

(a) For what values of $ k $ does the function $ y = \cos kt $ satisfy the differential equation $ 4y^{"} = - 25y? $ (b) For those values of $ k. $ verify that every member of the family functions $ y = A \sin kt + B \cos kt $ is also a solution.

Example 5

Which of the following functions are solutions of the differential equation $ y^{"} + y = \sin x ? $ (a) $ y = \sin x $ (b) $ y = \cos x $ (c) $ y = \frac {1}{2} x \sin x $ (d) $ y = - \frac{1}{2} x \cos x $
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Step-by-Step Explanations

QUESTION

Given the differential equation dP/dt = kP, how do you derive the general solution?

STEP-BY-STEP ANSWER:

Step 1: Separate variables by writing (1/P) dP = k dt.
Step 2: Integrate both sides to obtain ∫(1/P)dP = ∫k dt, which gives ln|P| = kt + C.
Step 3: Exponentiate both sides to solve for P, yielding P = Ce^(kt), where C is an arbitrary constant.
Final Answer: The general solution is P(t) = Ce^(kt).

Exponential Growth Model

QUESTION

How do you obtain the general solution for a spring-mass system modeled by m d²x/dt² = -2kx?

STEP-BY-STEP ANSWER:

Step 1: Express the equation in standard form: d²x/dt² = - (2k/m)x.
Step 2: Recognize that the standard solution involves functions whose second derivative is proportional to the negative of the function; these are sine and cosine functions.
Step 3: Write the general solution as x(t) = A cos(√(2k/m) t) + B sin(√(2k/m) t), where A and B are constants determined by initial conditions.
Final Answer: The general solution is x(t) = A cos(√(2k/m) t) + B sin(√(2k/m) t).

Spring Motion Model (Second-Order Differential Equation)

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Common Mistakes

  • Failing to properly separate variables when solving first-order differential equations.
  • Ignoring the limitations of a model, such as assuming unlimited growth without considering carrying capacity in logistic models.
  • Confusing equilibrium solutions with trivial or zero solutions.
  • Overlooking the role of initial conditions in determining a unique solution from a family of solutions.
  • Assuming that every differential equation has an easily obtainable explicit solution, rather than using graphical or numerical methods when necessary.