Book cover for Thomas Calculus

Thomas Calculus

George B. Thomas, Jr.

ISBN #9780321878960

13th Edition

6,812 Questions

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127,035 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces first-order differential equations and their solutions. It covers two major types: separable equations and linear equations. Key techniques include the use of slope fields to visualize solution behavior and Euler’s method for constructing numerical approximations. The discussion also emphasizes the distinction between general and particular solutions and prepares the groundwork for understanding more advanced topics in differential equations.

Learning Objectives

1

Explain the concept and formulation of first-order differential equations and distinguish between general and particular solutions.

2

Describe the role of slope fields in visualizing solution curves of differential equations.

3

Apply Euler’s method to obtain numerical approximations for initial value problems.

4

Identify the standard form of first-order linear differential equations and the significance of functions P(x) and Q(x).

Key Concepts

CONCEPT

DEFINITION

First-Order Differential Equation

An equation of the form dy/dx = f(x, y) that involves only the first derivative of the unknown function y(x).

General Solution

A solution containing an arbitrary constant that represents the entire family of solutions to a differential equation.

Particular Solution

A solution that satisfies a given initial condition, often represented by y(x0) = y0.

Initial Value Problem

A differential equation accompanied by an initial condition which the solution must satisfy.

Slope Field (Direction Field)

A graphical representation consisting of short line segments whose slopes correspond to the value of f(x, y) at various points, illustrating the overall behavior of solution curves.

Euler’s Method

A numerical technique that uses the linearization L(x) = y0 + f(x0, y0)(x - x0) to approximate solutions by taking small steps along the curve.

Linear Differential Equation

An equation of the form dy/dx + P(x)y = Q(x), where y and its derivative appear only to the first power and are not multiplied together.

Example Problems

Example 1

In Exercises $1-4,$ match the differential equations with their slope fields, graphed here. $$ y^{\prime}=x+y $$

Example 2

In Exercises $1-4,$ match the differential equations with their slope fields, graphed here. $$ y^{\prime}=y+1 $$

Example 3

In Exercises $1-4,$ match the differential equations with their slope fields, graphed here. $$ y^{\prime}=-\frac{x}{y} $$

Example 4

In Exercises $1-4,$ match the differential equations with their slope fields, graphed here. $$ y^{\prime}=y^{2}-x^{2} $$

Example 5

In Exercises 5 and $6,$ copy the slope fields and sketch in some of the solution curves. $$ y^{\prime}=(y+2)(y-2) $$
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Step-by-Step Explanations

QUESTION

Approximate the solution of the initial value problem y' = 1 + y, y(0) = 1 using Euler’s method with a step size dx = 0.1. Find y1, y2, and y3.

STEP-BY-STEP ANSWER:

Step 1: Start with the initial condition (x0, y0) = (0, 1).
Step 2: Compute the first approximation: x1 = x0 + dx = 0.1. Then use y1 = y0 + f(x0, y0) * dx. Since f(x, y) = 1 + y, substitute y0 to get y1 = 1 + (1 + 1)*0.1 = 1 + 0.2 = 1.2.
Step 3: Compute the second approximation: x2 = x1 + dx = 0.2. Now y2 = y1 + f(x1, y1) * dx = 1.2 + (1 + 1.2)*0.1 = 1.2 + 0.22 = 1.42.
Step 4: Compute the third approximation: x3 = x2 + dx = 0.3. Then y3 = y2 + f(x2, y2) * dx = 1.42 + (1 + 1.42)*0.1 = 1.42 + 0.242 = 1.662.
Final Answer: y1 ≈ 1.2, y2 ≈ 1.42, y3 ≈ 1.662.

Using Euler’s Method

QUESTION

Transform the differential equation x(dy/dx) = x² + 3y, for x > 0, into standard form.

STEP-BY-STEP ANSWER:

Step 1: Divide the entire equation by x (since x > 0) to isolate dy/dx.
Step 2: The equation becomes dy/dx = x + (3/x)y.
Step 3: Rewrite it in standard linear form: dy/dx - (3/x)y = x.
Final Answer: The standard form is dy/dx - (3/x)y = x, with P(x) = -3/x and Q(x) = x.

Rewriting a Differential Equation in Standard Form

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Common Mistakes

  • Neglecting to confirm that a candidate solution satisfies both the differential equation and the initial condition.
  • Mixing up the roles of general and particular solutions, particularly regarding the arbitrary constant.
  • Misinterpreting the standard form of linear differential equations by ignoring the sign associated with P(x).
  • Using a step size in Euler’s method that is too large, leading to significant accumulated error.