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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 9.4.15
Chapter 9, Problem 9.4.15

11–16. Initial value problems Solve the following initial value problems.


y'(t) − 3y = 12, y(1) = 4

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Identify the type of differential equation: This is a first-order linear differential equation of the form \(y'(t) + p(t)y = q(t)\), where \(p(t) = -3\) and \(q(t) = 12\).
Find the integrating factor \(\mu(t)\) using the formula \(\mu(t) = e^{\int p(t) \, dt}\). Here, calculate \(\mu(t) = e^{\int -3 \, dt}\).
Multiply both sides of the differential equation by the integrating factor \(\mu(t)\) to rewrite the left side as the derivative of a product: \(\frac{d}{dt}[\mu(t) y(t)] = \mu(t) q(t)\).
Integrate both sides with respect to \(t\) to find \(\mu(t) y(t) = \int \mu(t) q(t) \, dt + C\), where \(C\) is the constant of integration.
Use the initial condition \(y(1) = 4\) to solve for the constant \(C\), then solve for \(y(t)\) by dividing both sides by \(\mu(t)\).

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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(t)y = q(t). They can be solved using an integrating factor, which simplifies the equation into an exact derivative, allowing integration to find the general solution.
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Classifying Differential Equations

Integrating Factor Method

This method involves multiplying the differential equation by an integrating factor, typically e^(∫p(t)dt), to rewrite the left side as the derivative of a product. This facilitates direct integration to solve for y(t).
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Euler's Method

Initial Value Problems (IVP)

An IVP specifies the value of the solution at a particular point, such as y(1) = 4. After finding the general solution, the initial condition is used to determine the unique constant, yielding a specific solution.
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Initial Value Problems
Related Practice
Textbook Question

45–46. Harvesting problems Consider the harvesting problem in Example 6.

If r = 0.05 and H = 500, for what values of p₀ is the amount of the resource decreasing? For what value of p₀ is the amount of the resource constant? If p₀ = 9000, when does the resource vanish?

Textbook Question

Does the function y(t) = 2t satisfy the differential equation y'''(t) + y'(t) = 2?

Textbook Question

11–16. Initial value problems Solve the following initial value problems.


y'(x) = −y + 2, y(0) = −2

Textbook Question

17–18. {Use of Tech} Designing logistic functions Use the method of Example 1 to find a logistic function that describes the following populations. Graph the population function.


The population increases from 50 to 60 in the first month and eventually levels off at 150.

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) = 3 + e⁻²ᵗ

Textbook Question

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

y'(t) = yeᵗ, y(0) = −1

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