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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 9.1.21
Chapter 9, Problem 9.1.21

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

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Recognize that the given differential equation is a first-order ordinary differential equation of the form \(y'(t) = f(t)\), where \(f(t) = 3 + e^{-2t}\).
To find the general solution, integrate both sides with respect to \(t\): \(\int y'(t) \, dt = \int (3 + e^{-2t}) \, dt\).
Split the integral on the right side into two separate integrals: \(\int 3 \, dt + \int e^{-2t} \, dt\).
Integrate each term: the integral of \(3\) with respect to \(t\) is \$3t\(, and the integral of \)e^{-2t}\( with respect to \)t$ is \(-\frac{1}{2} e^{-2t}\) (using substitution or recognizing the exponential integral).
Combine the results and add the arbitrary constant of integration \(C\) to write the general solution: \(y(t) = 3t - \frac{1}{2} e^{-2t} + C\).

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Key Concepts

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

General Solution of a Differential Equation

The general solution of a differential equation includes all possible solutions and typically contains arbitrary constants. It represents the family of functions that satisfy the equation, capturing both particular and homogeneous parts.
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Integration of Functions

Solving first-order differential equations often involves integrating the right-hand side with respect to the independent variable. Integration reverses differentiation and helps find the original function from its derivative.
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Integrals of General Exponential Functions

Exponential Functions and Their Integrals

Exponential functions like e^(-2t) have specific integral formulas. For example, ∫e^(kt) dt = (1/k)e^(kt) + C, where k is a constant. Recognizing and integrating these correctly is essential for solving differential equations involving exponentials.
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Related Practice
Textbook Question

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

p'(x) = 2/(x² + x), p(1) = 0

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

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


y'(t) − 3y = 12, y(1) = 4

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

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

y'(t) = eᵗʸ, y(0) = 1

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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