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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 9.1.51c
Chapter 9, Problem 9.1.51c

Another second-order equation Consider the differential equation y''(t) + k²y(t) = 0, where k is a positive real number.
c. Give the general solution of the equation for arbitrary k > 0 and verify your conjecture.

Verified step by step guidance
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Recognize that the given differential equation is a second-order linear homogeneous differential equation with constant coefficients: \(y''(t) + k^{2} y(t) = 0\), where \(k > 0\).
Write the characteristic equation associated with the differential equation by replacing \(y(t)\) with \(e^{\lambda t}\), which gives: \(\lambda^{2} + k^{2} = 0\).
Solve the characteristic equation for \(\lambda\): \(\lambda^{2} = -k^{2}\), so \(\lambda = \pm i k\), where \(i\) is the imaginary unit.
Since the roots are purely imaginary, the general solution to the differential equation is a linear combination of sine and cosine functions: \(y(t) = C_{1} \cos(k t) + C_{2} \sin(k t)\), where \(C_{1}\) and \(C_{2}\) are arbitrary constants.
Verify the solution by differentiating \(y(t)\) twice to find \(y''(t)\), then substitute \(y(t)\) and \(y''(t)\) back into the original equation \(y''(t) + k^{2} y(t) = 0\) to confirm that it holds true for all \(t\).

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Key Concepts

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

Second-Order Linear Homogeneous Differential Equations

These are differential equations involving the second derivative of a function, with no terms independent of the function or its derivatives. The general form is y'' + p(t)y' + q(t)y = 0. Solutions depend on the characteristic equation derived from constant coefficients, which determines the behavior of the solution.
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Classifying Differential Equations

Characteristic Equation and Roots

For constant coefficient equations like y'' + k²y = 0, the characteristic equation is r² + k² = 0. Solving this yields complex roots ±ik, indicating oscillatory solutions. The nature of these roots guides the form of the general solution using sine and cosine functions.
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Root Test

General Solution of Homogeneous Equations with Complex Roots

When the characteristic roots are complex conjugates α ± βi, the general solution is y(t) = e^{αt}(C₁ cos(βt) + C₂ sin(βt)). For purely imaginary roots (α=0), the solution simplifies to y(t) = C₁ cos(kt) + C₂ sin(kt), representing oscillations with frequency k.
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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.

c. Sketch the solution curve that corresponds to the initial condition y0=1. 


y′(t) = 6 - 2y

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Textbook Question

29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.


c. Which time step results in the more accurate approximation? Explain your observations.


y′(t) = 4−y, y(0) = 3; y(t) = 4−e⁻ᵗ

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Textbook Question

27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.

c. Find the equilibrium points for the system.


x′(t) = −3x + xy, y′(t) = 2y − xy

Textbook Question

27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.


c. Find the equilibrium points for the system.


x′(t) = −3x + 6xy, y′(t) = y − 4xy

Textbook Question

Solving Bernoulli equations Use the method outlined in Exercise 43 to solve the following Bernoulli equations.


c. y′(t) + y = √y

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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.


c. Why is the condition A < T₀/2 needed?

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