Book cover for Calculus with Applications

Calculus with Applications

Margaret L. Lial • Raymond N. Greenwell • Nathan P. Ritchey

ISBN #9781292108971

11th Edition

3,612 Questions

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224,424 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces differential equations with an emphasis on elementary and separable types. Students learn to solve these equations by finding antiderivatives and using separation of variables, while also exploring real-world applications such as population growth, interest rate forecasting, and logistic growth. Key insights include understanding general versus particular solutions, the importance of initial conditions, and the concept of equilibrium points and their stability in dynamic models.

Learning Objectives

1

Explain what a differential equation is and distinguish between general and particular solutions.

2

Solve elementary differential equations using antiderivatives and integration techniques.

3

Apply the separation of variables method to solve separable differential equations.

4

Model real-world phenomena such as interest rates, population growth, and logistic growth using differential equations.

5

Analyze equilibrium points and determine their stability in autonomous differential equations.

Key Concepts

CONCEPT

DEFINITION

Differential Equation

An equation that involves an unknown function and a finite number of its derivatives.

General Solution

The complete family of solutions of a differential equation including an arbitrary constant (or constants).

Particular Solution

A specific solution of a differential equation that satisfies a given initial condition.

Separable Differential Equation

A differential equation that can be written in the form dy/dx = f(x) where the variables can be separated so that all y terms (with dy) are on one side and all x terms (with dx) are on the other side before integrating.

Logistic Equation

A model for growth in which the rate of increase is proportional both to the current quantity and to the difference between the current quantity and a maximum carrying capacity, usually written as dy/dt = k * y * (1 - y/N) with solution y = N/(1 + be^(-kt)).

Equilibrium Point

A constant solution to an autonomous differential equation where the derivative equals zero; its stability is determined by the behavior of solutions nearby.

Example Problems

Example 1

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation. $$\frac{d y}{d x}=-2 x+9 x^{2}$$

Example 2

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation. $$\frac{d y}{d x}=-4 x+12 x^{2}$$

Example 3

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation. $$4 x^{3}-2 \frac{d y}{d x}=0$$

Example 4

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation. $$3 x^{2}-3 \frac{d y}{d x}=2$$

Example 5

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation. $$7 y \frac{d y}{d x}=5 x^{2}$$
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Step-by-Step Explanations

QUESTION

Find the general solution of the differential equation dy/dx = 3x² - 2x.

STEP-BY-STEP ANSWER:

Step 1: Recognize that y must be an antiderivative of 3x² - 2x.
Step 2: Integrate the right-hand side with respect to x: ∫(3x² - 2x) dx.
Step 3: Compute the integral: ∫3x² dx = x³ and ∫2x dx = x², so the antiderivative is x³ - x².
Step 4: Add the constant of integration, C, to obtain the general solution: y = x³ - x² + C.
Final Answer: y = x³ - x² + C.

Solving dy/dx = 3x² - 2x

QUESTION

Solve the differential equation y (dy/dx) = x² by separation of variables.

STEP-BY-STEP ANSWER:

Step 1: Write the equation in separable form: y dy = x² dx.
Step 2: Integrate both sides: ∫ y dy = ∫ x² dx.
Step 3: Evaluate the integrals: (1/2)y² = (1/3)x³ + C.
Step 4: Multiply both sides by 2 (or leave as is) to write the solution implicitly: y² = (2/3)x³ + C'.
Final Answer: y² = (2/3)x³ + C, where C is an arbitrary constant.

Separation of Variables

QUESTION

For a goat population with a carrying capacity N, initial population yâ‚€, and growth constant k, derive the logistic equation solution.

STEP-BY-STEP ANSWER:

Step 1: Start with the logistic differential equation: dy/dt = k y (1 - y/N).
Step 2: Separate variables so that terms in y and dy are on one side and terms in t and dt are on the other.
Step 3: Integrate both sides using partial fractions.
Step 4: After integration, the solution can be written in implicit form, then solved explicitly to yield: y = N / (1 + b e^(-kt)), where b = (N - yâ‚€) / yâ‚€.
Final Answer: y = N / (1 + ((N - yâ‚€)/yâ‚€)e^(-kt)).

Logistic Growth Model

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Common Mistakes

  • Failing to include the constant of integration when computing antiderivatives.
  • Attempting to integrate non-separable equations without first manipulating them into separable form.
  • Confusing general solutions with particular solutions, especially when initial conditions are provided.
  • Misinterpreting the derivative dy/dx as a simple fraction, which can lead to errors when separating variables.
  • Not checking for any lost solutions when dividing by expressions that could be zero in separation of variables.