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

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Textbook Question

5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.

y'(t) = eʸᐟ²sin t

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Textbook Question

33–38. {Use of Tech} Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one function, be sure to indicate which function corresponds to the solution of the initial value problem.

yy'(x) = 2x/(2 + y)², y(1) = −1

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Textbook Question

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

y'(t) = 1 + eᵗ, y(0) = 4

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Textbook Question

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

y(t) = sec² t/(2y), y(π/4) = 1

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Textbook Question

12–16. Sketching direction fields Use the window [-2, 2] x [-2, 2] to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed.

y'(t) = 4−y, y(0) = −1

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