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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 9 - Differential Equations문제 9.1.51c
9장, 문제 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.

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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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
07:39
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.
추천 영상:
가이드 코스
07:15
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.
추천 영상:
04:00
Solutions to Basic Differential Equations
관련 실천
교과서 질문

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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교과서 질문

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.


(Use of Tech) Chemical rate equations The reaction of certain chemical compounds can be modeled using a differential equation of the form y'(t) = -kyⁿ(t), where y(t) is the concentration of the compound, for t ≥ 0, k > 0 is a constant that determines the speed of the reaction, and n is a positive integer called the order of the reaction. Assume the initial concentration of the compound is y(0) = y₀ > 0.


c. Let y₀ = 1 and k = 0.1. Graph the first-order and second-order solutions found in parts (a) and (b). Compare the two reactions. 

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교과서 질문

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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교과서 질문

{Use of Tech} Fish harvesting A fish hatchery has 500 fish at t=0, when harvesting begins at a rate of b>0fish/year The fish population is modeled by the initial value problem y′(t)=0.01y−b,y(0)=500 where t is measured in years.


c. Graph the solution in the case that b=60fish/year. Describe the solution.

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교과서 질문

[Use of Tech] Analysis of a separable equation Consider the differential equation yy'(t) = ½eᵗ + t and carry out the following analysis.

c. Graph the solutions in part (b) and describe their behavior as t increases. 

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교과서 질문

{Use of Tech} Free fall An object in free fall may be modeled by assuming the only forces at work are the gravitational force and air resistance. By Newton’s Second Law of Motion (mass end . acceleration = the sum of external forces), the velocity of the object satisfies the differential equation 


m · v'(t) = mg + f(v)

mass | acceleration | external forces


where f is a function that models the air resistance (assuming the positive direction is downward). One common assumption (often used for motion in air) is that f(v)=−kv^2, for t≥0, where k>0 is a drag coefficient.


c. Find the solution of this separable equation assuming v(0)=0 and 0<v²<g/a. 

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