Book cover for Calculus and Its Applications

Calculus and Its Applications

Larry J. Goldstein, David C. Lay, David I. Schneider

ISBN #9780134437774

14th Edition

3,033 Questions

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63,186 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces differential equations, highlighting their role in modeling dynamic systems across various fields. It covers essential methods for solving these equations including separation of variables and integrating factors for first-order linear equations, along with techniques for graphing solutions and employing numerical methods. The understanding of these methods enables the application of differential equations to both theoretical analyses and practical real-world problems.

Learning Objectives

1

Explain the concept and significance of differential equations and their solutions.

2

Solve differential equations using methods such as separation of variables and integrating factors.

3

Analyze and solve first-order linear differential equations and apply them to real-world problems.

4

Graph the solutions of differential equations to understand their behavior over time.

5

Explore numerical methods for approximating solutions when analytical methods are not applicable.

Key Concepts

CONCEPT

DEFINITION

Differential Equation

An equation that involves derivatives of an unknown function with respect to one or more variables. It models changes in physical, biological, and economic systems.

Solution

A function that, when substituted into the differential equation, satisfies the equation. This solution may be general or particular depending on constants and initial conditions.

Separation of Variables

A method for solving differential equations by rewriting the equation so that each variable appears on a different side before integrating both sides.

First-Order Linear Differential Equation

A differential equation of the form y' + p(t)y = q(t). It is solved by multiplying through by an integrating factor, which simplifies the left-hand side to the derivative of a product.

Numerical Methods

Techniques, such as Euler's method, used to approximate solutions of differential equations when analytical solutions are difficult or impossible to obtain.

Graphing Solutions

The process of plotting the solution curves of differential equations on a graph to visualize how the solution behaves over the independent variable (often time).

Example Problems

Example 1

Show that the function $f(t)=\frac{3}{2} e^{t^{2}}-\frac{1}{2}$ is a solution of the differential equation $y^{\prime}-2 t y=t$.

Example 2

Show that the function $f(t)=t^{2}-\frac{1}{2}$ is a solution of the differential equation $\left(y^{\prime}\right)^{2}-4 y=2$.

Example 3

Show that the function $f(t)=5 e^{2 t}$ satisfies $y^{\prime \prime}-3 y^{\prime}+2 y=0$ $y(0)=5, y^{\prime}(0)=10$.

Example 4

Show that the function $f(t)=\left(e^{-t}+1\right)^{-1}$ satisfies $y^{\prime}+y^{2}=y$ $y(0)=\frac{1}{2}$.

Example 5

State the order of the differential equation and verify that the given function is a solution. $$\left(1-t^{2}\right) y^{\prime \prime}-2 t y^{\prime}+2 y=0, y(t)=t$$
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Step-by-Step Explanations

QUESTION

Given the separable differential equation dy/dt = 6t + 3, solve for y(t).

STEP-BY-STEP ANSWER:

Step 1: Recognize that the equation is separable since the right-hand side depends only on t.
Step 2: Write the equation in differential form: dy = (6t + 3) dt.
Step 3: Integrate both sides: ∫ dy = ∫ (6t + 3) dt.
Step 4: Perform the integration: y = 3t² + 3t + C, where C is the constant of integration.
Final Answer: y(t) = 3t² + 3t + C.

Separation of Variables

QUESTION

Solve the differential equation y' + 3y = -t using an integrating factor.

STEP-BY-STEP ANSWER:

Step 1: Identify the equation in standard form: y' + 3y = -t, where p(t) = 3 and q(t) = -t.
Step 2: Compute the integrating factor μ(t) = e^(∫3 dt) = e^(3t).
Step 3: Multiply the entire equation by e^(3t) to obtain: e^(3t)y' + 3e^(3t)y = -t e^(3t).
Step 4: Recognize that the left-hand side is the derivative of e^(3t)y, i.e., d/dt [e^(3t)y] = -t e^(3t).
Step 5: Integrate both sides with respect to t: ∫ d/dt [e^(3t)y] dt = ∫ (-t e^(3t)) dt.
Step 6: Solve the right-hand integral using integration by parts.
Step 7: Divide by e^(3t) to solve for y(t) and include the constant of integration.
Final Answer: The general solution is y(t) = (Resulting function from integration)/e^(3t), with the constant of integration included.

First-Order Linear Differential Equation

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Common Mistakes

  • Failing to properly separate variables, leading to incorrect integration limits.
  • Forgetting the constant of integration after performing indefinite integrals.
  • Miscomputing the integrating factor by neglecting proper integration of p(t).
  • Overlooking initial or boundary conditions necessary to determine a unique particular solution.