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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 9.4.10
Chapter 9, Problem 9.4.10

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


v'(y) − v/2 = 14

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Identify the type of differential equation given: it is a first-order linear ordinary differential equation of the form \(v'(y) + P(y)v = Q(y)\), where \(P(y) = -\frac{1}{2}\) and \(Q(y) = 14\).
Find the integrating factor \(\mu(y)\) using the formula \(\mu(y) = e^{\int P(y) \, dy}\). In this case, calculate \(\mu(y) = e^{\int -\frac{1}{2} \, dy}\).
Multiply both sides of the differential equation by the integrating factor \(\mu(y)\) to rewrite the left side as the derivative of the product \(\mu(y) v(y)\).
Integrate both sides with respect to \(y\) to find \(\mu(y) v(y) = \int \mu(y) Q(y) \, dy + C\), where \(C\) is the constant of integration.
Solve for \(v(y)\) by dividing both sides by \(\mu(y)\) to express the general solution explicitly.

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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

These are differential equations of the form y' + P(x)y = Q(x), where the highest derivative is first order and the equation is linear in the unknown function and its derivative. Solving them typically involves finding an integrating factor to simplify the equation.
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Classifying Differential Equations

Integrating Factor Method

This method involves multiplying the entire differential equation by an integrating factor, usually e^(∫P(x)dx), which transforms the left side into the derivative of a product. This allows the equation to be integrated easily to find the general solution.
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Euler's Method

General Solution of Differential Equations

The general solution includes all possible solutions of a differential equation and typically contains an arbitrary constant. It represents the family of functions satisfying the equation, encompassing both particular and homogeneous solutions.
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Solutions to Basic Differential Equations
Related Practice
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