Book cover for Calculus of a Single Variable

Calculus of a Single Variable

Ron Larson, Bruce Edwards

ISBN #9781285060286

10th Edition

6,214 Questions

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101,749 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers essential methods for solving differential equations, ranging from finding general and particular solutions using initial conditions to graphical and numerical techniques such as slope fields and Euler’s Method. Students learn to verify solutions by substitution, sketch approximate solution curves from slope fields, and use separation of variables to solve more general differential equations. The emphasis on understanding the behavior of solution curves both analytically and visually equips learners to model real-world phenomena effectively.

Learning Objectives

1

Explain and differentiate between general and particular solutions of differential equations.

2

Use initial conditions to find particular solutions from the general solution of a differential equation.

3

Interpret and sketch slope fields to visualize solution curves for differential equations.

4

Apply Euler’s Method to approximate solutions numerically and understand the trade-off between step size and accuracy.

5

Utilize separation of variables as a strategy for solving more general differential equations.

Key Concepts

CONCEPT

DEFINITION

Differential Equation

An equation involving an unknown function and its derivatives. It describes how the function changes and can model physical phenomena such as growth, decay, or motion.

General Solution

The family of all solutions to a differential equation, usually containing one or more arbitrary constants.

Particular Solution

A specific solution obtained by applying initial conditions to the general solution, thereby determining the arbitrary constants.

Slope Field (Direction Field)

A graphical representation of a differential equation where short line segments at various points indicate the slope given by the differential equation at that point.

Euler’s Method

A numerical technique for approximating solutions of differential equations by stepping forward a small increment using the tangent slope at each point.

Separation of Variables

A technique for solving differential equations by rewriting them so that each variable and its differential are on opposite sides of the equation, allowing integration.

Example Problems

Example 1

Verifying a Solution In Exercises $1-8,$ verify the solution of the differential equation. $$ y=C e^{4 x} \quad y^{\prime}=4 y $$

Example 2

Verifying a Solution In Exercises $1-8,$ verify the solution of the differential equation. $$ y=e^{-2 x} \quad 3 y^{\prime}+5 y=-e^{-2 x} $$

Example 3

Verifying a Solution In Exercises $1-8,$ verify the solution of the differential equation. $$ x^{2}+y^{2}=C y \quad y^{\prime}=\frac{2 x y}{x^{2}-y^{2}} $$

Example 4

Verifying a Solution In Exercises $1-8,$ verify the solution of the differential equation. $$ y^{2}-2 \ln y=x^{2} \quad \frac{d y}{d x}=\frac{x y}{y^{2}-1} $$

Example 5

Verifying a Solution In Exercises $1-8,$ verify the solution of the differential equation. $$ y=C_{1} \sin x-C_{2} \cos x \quad y^{\prime \prime}+y=0 $$
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Step-by-Step Explanations

QUESTION

Given a differential equation y' = 2x - y and a candidate solution y = Ce^(x) + x^2, how do you verify if this function is a solution?

STEP-BY-STEP ANSWER:

Step 1: Differentiate the candidate solution y = Ce^(x) + x^2 to get y' = Ce^(x) + 2x.
Step 2: Substitute y and y' into the differential equation: Check if Ce^(x) + 2x equals 2x - (Ce^(x) + x^2).
Step 3: Simplify the expression to see if both sides match for some constant C. If they do not match, the candidate is not a solution.
Final Answer: Verification is done by ensuring that the left-hand side (computed derivative) equals the right-hand side (expression formed by the candidate solution) for all x in the domain.

Verifying a Solution

QUESTION

How do you approximate the solution of the differential equation y' = x - y with initial condition y(0) = 1 using Euler’s Method with a step size h = 0.1?

STEP-BY-STEP ANSWER:

Step 1: Start with the initial condition (x0, y0) = (0, 1).
Step 2: Compute the slope at (0,1): F(0,1) = 0 - 1 = -1.
Step 3: Calculate the next value using y1 = y0 + h * F(x0, y0) = 1 + 0.1 * (-1) = 0.9 and update x1 = x0 + 0.1 = 0.1.
Step 4: Repeat the process: Compute F(0.1, 0.9), then y2 = y1 + h * F(0.1,0.9), and so on.
Final Answer: The table of approximations is built step-by-step by updating x and y using the formula y(n) = y(n-1) + h * F(x(n-1), y(n-1)).

Using Euler’s Method

QUESTION

How do you solve the differential equation dy/dx = (2x)/(y) by separation of variables?

STEP-BY-STEP ANSWER:

Step 1: Rewrite the differential equation so that all y terms are on one side and x terms are on the other: y dy = 2x dx.
Step 2: Integrate both sides: ∫ y dy = ∫ 2x dx.
Step 3: Perform the integrations: (1/2)y^2 = x^2 + C.
Step 4: Solve for y if needed, or leave the answer in implicit form.
Final Answer: The general solution of the differential equation is given by (1/2)y^2 = x^2 + C.

Separation of Variables

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Common Mistakes

  • Failing to properly apply the initial conditions to determine the arbitrary constants, leading to incorrect particular solutions.
  • Mixing up or misinterpreting the roles of the general and particular solutions.
  • Misreading slopes in a slope field, which can result in inaccurate sketches of the solution curves.
  • Using a step size in Euler’s Method that is too large, thereby increasing the numerical error and reducing the accuracy of the approximation.
  • Incorrectly separating the variables; for example, failing to move all instances of one variable to one side of the equation before integration.