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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.R.3
Chapter 9, Problem 9.R.3

2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.
y′(t) + 2y = 6

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1
Identify the type of differential equation. The given equation \(y'(t) + 2y = 6\) is a first-order linear ordinary differential equation.
Rewrite the equation in the standard linear form: \(y' + P(t)y = Q(t)\), where \(P(t) = 2\) and \(Q(t) = 6\).
Find the integrating factor (IF) using the formula \(\mu(t) = e^{\int P(t)\,dt}\). Here, calculate \(\mu(t) = e^{\int 2\,dt} = e^{2t}\).
Multiply both sides of the differential equation by the integrating factor \(e^{2t}\) to get \(e^{2t}y' + 2e^{2t}y = 6e^{2t}\). Notice that the left side is the derivative of \(e^{2t}y\).
Integrate both sides with respect to \(t\): \(\int \frac{d}{dt}(e^{2t}y)\,dt = \int 6e^{2t}\,dt\). Then solve for \(y(t)\) by dividing by \(e^{2t}\) and include 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

These are differential equations of the form y' + p(t)y = q(t), where y' is the derivative of y with respect to t. They can be solved using integrating factors or other methods to find the general solution that satisfies the equation for all t.
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Classifying Differential Equations

Integrating Factor Method

This technique involves multiplying the entire differential equation by an integrating factor, usually e^(∫p(t)dt), to rewrite the left side as the derivative of a product. This simplifies solving the equation by allowing direct integration to find y(t).
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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 that satisfy the equation, encompassing both particular and homogeneous solutions.
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Solutions to Basic Differential Equations
Related Practice
Textbook Question

Direction fields Consider the direction field for the equation y′=y(2−y) shown in the figure and initial conditions of the form y(0)=A.

a. Sketch a solution on the direction field with the initial condition y(0)=1.

Textbook Question

Euler’s metho d Consider the initial value problem y′(t)=1/2y,y(0)=1. 

a. Use Euler’s method with Δt=0.1 to compute approximations to y(0.1) and y(0.2). 

Textbook Question

Logistic growth parameters A cell culture has a population of 20 when a nutrient solution is added at t=0. After 20 hours, the cell population is 80 and the carrying capacity of the culture is estimated to be 1600 cells.

c. After how many hours does the population reach half of the carrying capacity

Textbook Question

22–25. Equilibrium solutions Find the equilibrium solutions of the following equations and determine whether each solution is stable or unstable.

y′(t) = y(3+y)(y-5)

Textbook Question

Logistic growth in India The population of India was 435 million in 1960 (t=0) and 487 million in 1965 (t=5). The projected population for 2050 is 1.57 billion.

b. Use the solution of the logistic equation and the 2050 projected population to determine the carrying capacity.

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Textbook Question

2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.

y′(t) = √(y/t)