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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 9.1.50a
Chapter 9, Problem 9.1.50a

A second-order equation Consider the differential equation y''(t) - k²y(t) = 0 where k > 0 is a real number.


a. Verify by substitution that when k = 1, a solution of the equation is y(t) = C₁eᵗ + C₂e⁻ᵗ. You may assume this function is the general solution.

Verified step by step guidance
1
Start by writing down the given differential equation for the case when \( k = 1 \): \[ y''(t) - y(t) = 0 \]
Write the proposed solution for \( k = 1 \): \[ y(t) = C_1 e^{t} + C_2 e^{-t} \]
Compute the first derivative \( y'(t) \) of the proposed solution: \[ y'(t) = C_1 e^{t} - C_2 e^{-t} \]
Compute the second derivative \( y''(t) \) of the proposed solution: \[ y''(t) = C_1 e^{t} + C_2 e^{-t} \]
Substitute \( y(t) \) and \( y''(t) \) back into the differential equation and simplify: \[ y''(t) - y(t) = (C_1 e^{t} + C_2 e^{-t}) - (C_1 e^{t} + C_2 e^{-t}) = 0 \] Since this holds true for all \( t \), the proposed function is indeed a solution.

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Key Concepts

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

Second-Order Linear Differential Equations

These are differential equations involving the second derivative of a function. They often have the form y'' + p(t)y' + q(t)y = 0. Solutions typically involve characteristic equations that help find general solutions based on roots.
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Classifying Differential Equations

Characteristic Equation and Its Roots

For constant-coefficient linear differential equations like y'' - k²y = 0, the characteristic equation is r² - k² = 0. Solving this quadratic gives roots that determine the form of the general solution, such as exponential functions when roots are real.
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Root Test

Verification by Substitution

To verify a proposed solution, substitute it and its derivatives into the original differential equation. If the equation holds true for all t, the function is a valid solution. This method confirms the correctness of the general solution.
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Substitution With an Extra Variable
Related Practice
Textbook Question

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

a. Find the equilibrium solutions. 


y′(t) = y(y - 3)

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Textbook Question

{Use of Tech} Logistic equation for spread of rumors Sociologists model the spread of rumors using logistic equations. The key assumption is that at any given time, a fraction y of the population, where 0≤y≤1, knows the rumor, while the remaining fraction 1−y does not. Furthermore, the rumor spreads by interactions between those who know the rumor and those who do not. The number of such interactions is proportional to y1−y. Therefore, the equation that describes the spread of the rumor is y′(t)=ky(1−y), for t≥0 where k is a positive real number and t is measured in weeks. The number of people who initially know the rumor is y(0)=y0, where 0≤y0≤1. 

a. Solve this initial value problem and give the solution in terms of k and y0.

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Textbook Question

U.S. population projections According to the U.S. Census Bureau, the nation’s population (to the nearest million) was 296 million in 2005 and 321 million in 2015. The Bureau also projects a 2050 population of 398 million. To construct a logistic model, both the growth rate and the carrying capacity must be estimated. There are several ways to estimate these parameters. Here is one approach:


a. Assume t = 0 corresponds to 2005 and that the population growth is exponential for the first ten years; that is, between 2005 and 2015, the population is given by P(t) = P(0)exp(rt). Estimate the growth rate r using this assumption.

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Textbook Question

Cooling time Suppose an object with an initial temperature of T₀ > 0 is put in surroundings with an ambient temperature of A, where A < T₀/2. Let t₁/₂ be the time required for the object to cool to T₀/2.


a. Show that t₁/₂ = −1/k ln((T₀ − 2A)/(2(T₀ − A))).

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Textbook Question

42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.

a. Find the general solution of the equation.


y'(t) = t²/(y² + 1); y(−1) = 1, y(0) = 0, y(−1) = −1


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Textbook Question

23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.

a. Write an initial value problem for the mass of the substance.


A 500-L tank is initially filled with pure water. A copper sulfate solution with a concentration of 20 g/L flows into the tank at a rate of 4 L/min. The thoroughly mixed solution is drained from the tank at a rate of 4 L/min.

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