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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 9 - Differential EquationsProblem 9.4.6
Chapter 9, Problem 9.4.6

5–10. First-order linear equations Find the general solution of the following equations.


y'(x) = −y + 2

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1
Rewrite the differential equation in the standard linear form: \(y'(x) + y = 2\).
Identify the integrating factor \(\mu(x)\), which is given by \(\mu(x) = e^{\int 1 \, dx} = e^{x}\).
Multiply both sides of the equation by the integrating factor to get: \(e^{x} y' + e^{x} y = 2 e^{x}\).
Recognize that the left side is the derivative of the product \(e^{x} y\), so write it as \(\frac{d}{dx} (e^{x} y) = 2 e^{x}\).
Integrate both sides with respect to \(x\): \(\int \frac{d}{dx} (e^{x} y) \, dx = \int 2 e^{x} \, dx\), then solve for \(y\) by dividing by \(e^{x}\) and adding the constant of integration.

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Key Concepts

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

First-Order Linear Differential Equations

A first-order linear differential equation has the form y' + p(x)y = q(x). It involves the first derivative of the unknown function and can be solved using integrating factors or other standard methods. Recognizing this form is essential to apply the correct solution technique.
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Integrating Factor Method

The integrating factor method involves multiplying the entire differential equation by a specially chosen function, usually e^(∫p(x)dx), to rewrite the left side as a derivative of a product. This simplifies solving the equation by allowing direct integration.
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General Solution of Differential Equations

The general solution includes all possible solutions of a differential equation and typically contains an arbitrary constant. It combines the homogeneous solution (solving y' + p(x)y = 0) and a particular solution to the nonhomogeneous equation.
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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) = y³sin t, y(0) = 1

Textbook Question

The general solution of a first-order linear differential equation is y(t) = Ce⁻¹⁰ᵗ − 13. What solution satisfies the initial condition y(0) = 4?

Textbook Question

21–24. Logistic equations Consider the following logistic equations. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed direction field is not needed. Assume t ≥ 0 and tP ≥ 0.

P′(t) = 0.05P(1−P/800); P(0) = 100, P(0) = 400, P(0) = 700

Textbook Question

21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.

y'(t) = t lnt + 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

Textbook Question

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

y'(x) = 4 sec² 2x, y(0) = 8