Explanation Given information : The given system of equations is d x d t =- x + 3 y d y d t =- 3 x - y The given solution is Y 1 ( t ) = ( e - t sin ( 3 t ) , e - t cos ( 3 t ) ) Concept used : If the vector Y ( x ( t ) , y ( t ) ) is the solution of the given system of equations if the point ( x ( t ) , y ( t ) ) = ( f 1 ( t ) , f 2 ( t ) ) lies on the curve represented by the system of equations. That satisfies the equations. Calculations: Thegiven system of equations may be written as d x ( t ) d t = - x ( t ) + 3 y ( t ) .... (1) d y ( t ) d t =-3 x ( t ) - y ( t ) .... (2) Given solution is Y 1 ( t ) = ( e - t sin ( 3 t ) , e - t cos ( 3 t ) ) , i.e. x ( t ) = e - t sin ( 3 t ) and y ( t ) = e - t cos ( 3 t ) . We verify that x ( t ) = e - t sin ( 3 t ) and y ( t ) = e - t cos ( 3 t ) satisfy the equation (1) and (2). Left-hand side of equation (1) : d x ( t ) d t = d ( e - t sin ( 3 t ) ) d t = e - t ⋅ 3 cos ( 3 t ) - e - t sin ( 3 t ) = e - t ( 3 cos ( 3 t ) - sin ( 3 t ) ) Right-hand side of equation (1): - x ( t ) + 3 y ( t ) = - e

Differential Equations (with DE Tools Printed Access Card)

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ISBN: 9781133109037

Author: Paul Blanchard

Publisher: Cengage Learning

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Chapter 2.6, Problem 3E

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Toverify: Y1(t)=(e- tsin(3t),e- tcos(3t)) is the solution of the given system of equations.

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Chapter 2.6, Problem 2E

Chapter 2.6, Problem 4E

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Chapter 2 Solutions

Differential Equations (with DE Tools Printed Access Card)

Ch. 2.1 - Exercises 1-6 refer to the following systems of...Ch. 2.1 - Exercises 1-6 refer to the following systems of...Ch. 2.1 - Exercises 1-6 refer to the following systems of...Ch. 2.1 - Exercises 1-6 refer to the following systems of...Ch. 2.1 - Prob. 5E Ch. 2.1 - Prob. 6E Ch. 2.1 - Consider the predator-prey system...Ch. 2.1 - Consider the predator-prey system dRdt=2R(1R...Ch. 2.1 - Exercises 9-14 refer to the predator-prey and the...Ch. 2.1 - Exercises 9-14 refer to the predator-prey and the...

Ch. 2.1 - Exercises 9-14 refer to the predator-prey and the...Ch. 2.1 - Prob. 12E Ch. 2.1 - Prob. 13E Ch. 2.1 - Exercises 9-14 refer to the predator-prey and the...Ch. 2.1 - Prob. 15E Ch. 2.1 - Consider the system of predator-prey equations...Ch. 2.1 - Pesticides that kill all insect species are not...Ch. 2.1 - Some predator species seldom capture healthy adult...Ch. 2.1 - Prob. 19E Ch. 2.1 - Consider the initial-value problem d2ydt2+kmy=0...Ch. 2.1 - A mass weighing 12 pounds stretches a spring 3...Ch. 2.1 - A mass weighing 4 pounds stretches a spring 4...Ch. 2.1 - Do the springs in an “extra firm’ mattress have a...Ch. 2.1 - Consider a vertical mass-spring system as shown in...Ch. 2.1 - Exercises 25—30 refer to a situation in which...Ch. 2.1 - Prob. 26E Ch. 2.1 - Prob. 27E Ch. 2.1 - Prob. 28E Ch. 2.1 - Prob. 29E Ch. 2.1 - Exercises 25—30 refer to a situation in which...Ch. 2.2 - Prob. 1E Ch. 2.2 - Prob. 2E Ch. 2.2 - Prob. 3E Ch. 2.2 - Prob. 4E Ch. 2.2 - Prob. 5E Ch. 2.2 - Prob. 6E Ch. 2.2 - Prob. 7E Ch. 2.2 - Convert the second-order differential equation 1...Ch. 2.2 - Prob. 9E Ch. 2.2 - Consider the system dxdt=2x+ydydt=2y and its...Ch. 2.2 - Eight systems of differential equations and four...Ch. 2.2 - Consider the modified predator-prey system...Ch. 2.2 - In Exercises 13—18. (a) find the equilibrium...Ch. 2.2 - Prob. 14E Ch. 2.2 - Prob. 15E Ch. 2.2 - In Exercises 13—18. (a) find the equilibrium...Ch. 2.2 - Prob. 17E Ch. 2.2 - In Exercises 13—18. (a) find the equilibrium...Ch. 2.2 - Prob. 19E Ch. 2.2 - Prob. 20E Ch. 2.2 - Consider the four solution curves in the phase...Ch. 2.2 - Prob. 22E Ch. 2.2 - Prob. 23E Ch. 2.2 - Prob. 24E Ch. 2.2 - Prob. 25E Ch. 2.2 - Prob. 26E Ch. 2.2 - Prob. 27E Ch. 2.3 - In Exercises 1—4, a harmonic oscillator equation...Ch. 2.3 - In Exercises 1—4, a harmonic oscillator equation...Ch. 2.3 - In Exercises 1—4, a harmonic oscillator equation...Ch. 2.3 - In Exercises 1—4, a harmonic oscillator equation...Ch. 2.3 - Prob. 5E Ch. 2.3 - In the damped harmonic oscillator, we assume that...Ch. 2.3 - Consider any damped harmonic oscillator equation...Ch. 2.3 - Consider any damped harmonic oscillator equation...Ch. 2.3 - In Exercises 9 and 10, we consider a mass sliding...Ch. 2.3 - In Exercises 9 and 10, we consider a mass sliding...Ch. 2.4 - In Exercises 1-4, we consider the system...Ch. 2.4 - In Exercises 1-4, we consider the system...Ch. 2.4 - In Exercises 1-4, we consider the system...Ch. 2.4 - In Exercises 1-4, we consider the system...Ch. 2.4 - In Exercises 5-12, we consider the partially...Ch. 2.4 - Prob. 6E Ch. 2.4 - In Exercises 5-12, we consider the partially...Ch. 2.4 - Prob. 8E Ch. 2.4 - In Exercises 5-12, we consider the partially...Ch. 2.4 - In Exercises 5-12, we consider the partially...Ch. 2.4 - Prob. 11E Ch. 2.4 - Prob. 12E Ch. 2.4 - Consider the partially decoupled system...Ch. 2.5 - Prob. 1E Ch. 2.5 - Prob. 2E Ch. 2.5 - Prob. 3E Ch. 2.5 - In Exercises 3—6, a system, an initial condition,...Ch. 2.5 - Prob. 5E Ch. 2.5 - Prob. 6E Ch. 2.5 - Using a computer or calculator, apply Euler’s...Ch. 2.5 - Prob. 8E Ch. 2.6 - Consider the system dxdt=x+ydydt=y (a) Show that...Ch. 2.6 - Prob. 2E Ch. 2.6 - Prob. 3E Ch. 2.6 - Prob. 4E Ch. 2.6 - Prob. 5E Ch. 2.6 - Prob. 6E Ch. 2.6 - Prob. 7E Ch. 2.6 - (a) Suppose Y1(t) is a solution of an autonomous...Ch. 2.6 - Prob. 9E Ch. 2.6 - Consider the system dxdt=2dydt=y2 (a) Calculate...Ch. 2.6 - Consider the system dxdt=2dydt=y2 Show that, for...Ch. 2.7 - Prob. 1E Ch. 2.7 - In the SIR model, we assume that everyone in the...Ch. 2.7 - Vaccines make it possible to prevent epidemics....Ch. 2.7 - Prob. 4E Ch. 2.7 - Prob. 5E Ch. 2.7 - One of the basic assumptions of the SIR model is...Ch. 2.7 - Prob. 7E Ch. 2.7 - Prob. 8E Ch. 2.7 - Prob. 9E Ch. 2.7 - Using =1.66 and the value of that you determined...Ch. 2.8 - Prob. 1E Ch. 2.8 - Prob. 2E Ch. 2.8 - Prob. 3E Ch. 2.8 - Prob. 4E Ch. 2.8 - Prob. 5E Ch. 2 - Prob. 1RE Ch. 2 - Short answer exercises: Exercises 1-14 focus on...Ch. 2 - Short answer exercises: Exercises 1-14 focus on...Ch. 2 - Short answer exercises: Exercises 1-14 focus on...Ch. 2 - Short answer exercises: Exercises 1-14 focus on...Ch. 2 - Short answer exercises: Exercises 1-14 focus on...Ch. 2 - Prob. 7RE Ch. 2 - Prob. 8RE Ch. 2 - Prob. 9RE Ch. 2 - Prob. 10RE Ch. 2 - Prob. 11RE Ch. 2 - Prob. 12RE Ch. 2 - Short answer exercises: Exercises 1-14 focus on...Ch. 2 - Prob. 14RE Ch. 2 - Prob. 15RE Ch. 2 - Prob. 16RE Ch. 2 - Prob. 17RE Ch. 2 - Prob. 18RE Ch. 2 - Prob. 19RE Ch. 2 - Prob. 20RE Ch. 2 - Prob. 21RE Ch. 2 - Prob. 22RE Ch. 2 - Prob. 23RE Ch. 2 - Prob. 24RE Ch. 2 - Prob. 25RE Ch. 2 - Prob. 26RE Ch. 2 - Prob. 27RE Ch. 2 - Prob. 28RE Ch. 2 - Prob. 29RE Ch. 2 - Prob. 30RE Ch. 2 - In Exercises 31-34, a solution curve in the...Ch. 2 - Prob. 32RE Ch. 2 - Prob. 33RE Ch. 2 - Prob. 34RE Ch. 2 - Consider the partially decoupled system...Ch. 2 - Consider the partially decoupled system...Ch. 2 - Prob. 37RE

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Toverify : Y 1 ( t ) = ( e - t sin ( 3 t ) , e - t cos ( 3 t ) ) is the solution of the given system of equations.

Solution Summary: The author explains how the given system of equations is solved if the vector Y(x),y(t)) is the solution.

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Toverify: Y1(t)=(e- tsin(3t),e- tcos(3t)) is the solution of the given system of equations.

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Chapter 2 Solutions