Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.3.1

Chapter 9, Problem 9.3.1

What is a separable first-order differential equation?

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A separable first-order differential equation is a type of differential equation that can be written in the form \(\frac{dy}{dx} = g(x)h(y)\), where the right-hand side is a product of a function of \(x\) and a function of \(y\).

The key idea is that the variables \(x\) and \(y\) can be separated on opposite sides of the equation, allowing us to rewrite it as \(\frac{1}{h(y)} dy = g(x) dx\).

Once separated, we integrate both sides with respect to their own variables: \(\int \frac{1}{h(y)} dy = \int g(x) dx\).

After performing the integrations, we obtain an implicit or explicit solution involving \(x\) and \(y\).

This method is useful because it transforms the differential equation into two simpler integrals, making it easier to solve.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

First-Order Differential Equation

A first-order differential equation involves the first derivative of an unknown function with respect to an independent variable. It expresses a relationship between the function and its rate of change, typically written as dy/dx = f(x, y).

Separable Differential Equation

A separable differential equation is one where the variables can be separated on opposite sides of the equation, allowing it to be written as g(y) dy = h(x) dx. This form enables integration of each side independently to find the solution.

Method of Separation of Variables

The method of separation of variables solves separable equations by isolating y terms with dy and x terms with dx, then integrating both sides. This technique transforms a differential equation into two integrals, simplifying the process of finding explicit solutions.

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Separation of Variables

Related Practice

Textbook Question

17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.

y'(t) = cos² y, y(1) = π/4

Textbook Question

33–42. Solving initial value problems Solve the following initial value problems.

y'(t) = 1 + eᵗ, y(0) = 4

Textbook Question

9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.

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23–26. Loan problems The following initial value problems model the payoff of a loan. In each case, solve the initial value problem, for t≥0 graph the solution, and determine the first month in which the loan balance is zero.

B′(t) = 0.005B − 500, B(0) = 50,000

Textbook Question

Orthogonal trajectories Use the method in Exercise 44 to find the orthogonal trajectories for the family of circles x² + y² = a²

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Textbook Question