Solution: Solving a System of Linear Differential Equations

Finding Age Distribution Vectors In Exercises 16, use the age transition matrix L and age distribution vector x1 to find the age distribution vectors x2 and x3. Then find a stable age distribution vector. L = [ 0 1 2 2 0], x1 = [ 10 10]

Finding Age Distribution Vectors In Exercises 16, use the age transition matrix L and age distribution vector x1 to find the age distribution vectors x2 and x3. Then find a stable age distribution vector. L = [ 0 1 16 4 0], x1 = [ 160 160]

Finding Age Distribution Vectors In Exercises 16, use the age transition matrix L and age distribution vector x1 to find the age distribution vectors x2 and x3. Then find a stable age distribution vector. L = [ 0 1 0 3 0 1 2 4 0 0 ] , x1 = [ 12 12 12]

Finding Age Distribution Vectors In Exercises 16, use the age transition matrix L and age distribution vector x1 to find the age distribution vectors x2 and x3. Then find a stable age distribution vector. L = [ 0 1 2 0 2 0 1 2 0 0 0 ] , x1 = [ 8 8 8 ]

Finding Age Distribution Vectors In Exercises 16, use the age transition matrix L and age distribution vector x1 to find the age distribution vectors x2 and x3. Then find a stable age distribution vector. L = [ 0 1 4 0 0 2 0 1 0 2 0 0 1 2 0 0 0 0 ] , x1 = [ 100 100 100 100]

Finding Age Distribution Vectors In Exercises 16, use the age transition matrix L and age distribution vector x1 to find the age distribution vectors x2 and x3. Then find a stable age distribution vector. L = [ 0 1 2 0 0 0 6 0 1 0 0 4 0 0 1 2 0 0 0 0 0 1 2 0 0 0 0 0 ] , x1 = [ 24 24 24 24 24]

Population Growth Model A population has the characteristics below. (a) A total of 75% of the population survives the first year. Of that 75%, 25% survives the second year. The maximum life span is 3 years. (b) The average number of offspring for each member of the population is 2 the first year, 4 the second year, and 2 the third year. The population now consists of 160 members in each of the three age classes. How many members will there be in each age class in 1 year? in 2 years?

Population Growth Model A population has the characteristics below. (a) A total of 80% of the population survives the first year. Of that 80%, 25% survives the second year. The maximum life span is 3 years. (b) The average number of offspring for each member of the population is 3 the first year, 6 the second year, and 3 the third year. The population now consists of 120 members in each of the three age classes. How many members will there be in each age class in 1 year? in 2 years?

Population Growth Model A population has the characteristics below. (a) A total of 60% of the population survives the first year. Of that 60%, 50% survives the second year. The maximum life span is 3 years. (b) The average number of offspring for each member of the population is 2 the first year, 5 the second year, and 2 the third year. The population now consists of 100 members in each of the three age classes. How many members will there be in each age class in 1 year? in 2 years?

Find the limit (if it exists) of Anx1 as n approaches infinity, where A = [ 0 1 2 2 0] and x1 = [ a a].

Solving a System of Linear Differential Equations In Exercises 1120, solve the system of first-order linear differential equations. y1 = y2 = 2y1 y2

Solving a System of Linear Differential Equations In Exercises 1120, solve the system of first-order linear differential equations. y1 = y2 = 5y1 4y2

Solving a System of Linear Differential Equations In Exercises 1120, solve the system of first-order linear differential equations. y1 = y2 = 4y1 1 2y2

Solving a System of Linear Differential Equations In Exercises 1120, solve the system of first-order linear differential equations. y1 = y2 = 1 2y1 1 8y2

Solving a System of Linear Differential Equations In Exercises 1120, solve the system of first-order linear differential equations. y1 = y2 = y3 = y1 6y2 y3

Solving a System of Linear Differential Equations In Exercises 1120, solve the system of first-order linear differential equations. y1 = y2 = y3 = 5y1 2y2 3y3

Solving a System of Linear Differential Equations In Exercises 1120, solve the system of first-order linear differential equations. y1 = y2 = y3 = 0.3y1 0.4y2 0.6y3

Solving a System of Linear Differential Equations In Exercises 1120, solve the system of first-order linear differential equations. y1 = y2 = y3 = 2 3y1 3 5y2 8y3

Solving a System of Linear Differential Equations In Exercises 1120, solve the system of first-order linear differential equations. y1 = y2 = y3 = y4 = 7y1 9y2 7y3 9y4

Solving a System of Linear Differential Equations In Exercises 1120, solve the system of first-order linear differential equations. . y1 = y2 = y3 = y4 = 0.1y1 7 4y2 2y3 5y4

Solving a System of Linear Differential Equations In Exercises 2128, solve the system of first-order linear differential equations. y1 = y2 = y1 4y2 2y2

Solving a System of Linear Differential Equations In Exercises 2128, solve the system of first-order linear differential equations. y1 = y2 = y1 4y2 2y1 + 8y2

Solving a System of Linear Differential Equations In Exercises 2128, solve the system of first-order linear differential equations. y1 = y2 = y1 + 2y2 2y1 + y2

Solving a System of Linear Differential Equations In Exercises 2128, solve the system of first-order linear differential equations. y1 = y2 = y1 2y1 + y2 4y2

Solving a System of Linear Differential Equations In Exercises 2128, solve the system of first-order linear differential equations. y1 = y2 = y3 = y1 2y2 2y2 + y3 + 4y3 3y3

Solving a System of Linear Differential Equations In Exercises 2128, solve the system of first-order linear differential equations. y1 = y2 = y3 = 2y1 + y2 y1 + y2 y1 + y3 + y3

Solving a System of Linear Differential Equations In Exercises 2128, solve the system of first-order linear differential equations. y1 = y2 = y3 = 4y1 3y2 4y2 + 5y3 10y3 4y3

Solving a System of Linear Differential Equations In Exercises 2128, solve the system of first-order linear differential equations. y1 = y2 = y3 = 2y1 + y3 3y2 + 4y3 y

Writing a System and Verifying the General Solution In Exercises 2932, write the system of first-order linear differential equations represented by the matrix equation y = Ay. Then verify the general solution. A = [ 1 0 1 1], y1 = C1et + C2tet y2 = C2et

Writing a System and Verifying the General Solution In Exercises 2932, write the system of first-order linear differential equations represented by the matrix equation y = Ay. Then verify the general solution. A = [ 1 1 1 1], y1 = C1et cos t + C2et sin t y2 = C2et cos t + C1et sin t

Writing a System and Verifying the General Solution In Exercises 2932, write the system of first-order linear differential equations represented by the matrix equation y = Ay. Then verify the general solution. A = [ 0 0 0 1 0 4 0 1 0 ] , y1 = y2 = y3 = C1 + C2 cos 2t + 2C3 cos 2t 4C2 cos 2t C3 sin 2t 2C2 sin 2t 4C3 sin 2t

Writing a System and Verifying the General Solution In Exercises 2932, write the system of first-order linear differential equations represented by the matrix equation y = Ay. Then verify the general solution. A = [ 0 0 1 1 0 3 0 1 3 ] , y1 = y2 = y3 = C1et + C2tet + (C1 + C2)et + (C2 + 2C3)tet + (C1 + 2C2 + 2C3)et + (C2 + 4C3)tet + C3t 2et C3t 2et C3t 2e

Finding the Matrix of a Quadratic Form In Exercises 3338, find the matrix A of the quadratic form associated with the equation. x2 + y2 4 = 0

Finding the Matrix of a Quadratic Form In Exercises 3338, find the matrix A of the quadratic form associated with the equation. x2 4xy + y2 4 = 0

Finding the Matrix of a Quadratic Form In Exercises 3338, find the matrix A of the quadratic form associated with the equation. 9x2 + 10xy 4y2 36 = 0

Finding the Matrix of a Quadratic Form In Exercises 3338, find the matrix A of the quadratic form associated with the equation. 12x2 5xy x + 2y 20 = 0

Finding the Matrix of a Quadratic Form In Exercises 3338, find the matrix A of the quadratic form associated with the equation. 10xy 10y2 + 4x 48 = 0

Finding the Matrix of a Quadratic Form In Exercises 3338, find the matrix A of the quadratic form associated with the equation. 16x2 4xy + 20y 2 72 = 0

Finding the Matrix of a Quadratic Form In Exercises 3944, find the matrix A of the quadratic form associated with the equation. Then find the eigenvalues of A and an orthogonal matrix P such that PTAP is diagonal. 2x2 3xy 2y2 + 10 = 0

Finding the Matrix of a Quadratic Form In Exercises 3944, find the matrix A of the quadratic form associated with the equation. Then find the eigenvalues of A and an orthogonal matrix P such that PTAP is diagonal. 5x2 2xy + 5y2 + 10x 17 = 0

Finding the Matrix of a Quadratic Form In Exercises 3944, find the matrix A of the quadratic form associated with the equation. Then find the eigenvalues of A and an orthogonal matrix P such that PTAP is diagonal. 13x2 + 63xy + 7y2 16 = 0

Finding the Matrix of a Quadratic Form In Exercises 3944, find the matrix A of the quadratic form associated with the equation. Then find the eigenvalues of A and an orthogonal matrix P such that PTAP is diagonal. 3x2 23xy + y 2 + 2x + 23y = 0

Finding the Matrix of a Quadratic Form In Exercises 3944, find the matrix A of the quadratic form associated with the equation. Then find the eigenvalues of A and an orthogonal matrix P such that PTAP is diagonal. 16x2 24xy + 9y2 60x 80y + 100 = 0

Finding the Matrix of a Quadratic Form In Exercises 3944, find the matrix A of the quadratic form associated with the equation. Then find the eigenvalues of A and an orthogonal matrix P such that PTAP is diagonal. 17x2 + 32xy 7y2 75 = 0

Rotation of a Conic In Exercises 4552, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. 13x2 8xy + 7y 2 45 = 0

Rotation of a Conic In Exercises 4552, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. x2 + 4xy + y2 9 = 0

Rotation of a Conic In Exercises 4552, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. 2x2 4xy + 5y 2 36 = 0

Rotation of a Conic In Exercises 4552, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. 7x2 + 32xy 17y2 50 = 0

Rotation of a Conic In Exercises 4552, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. 2x2 + 4xy + 2y 2 + 62x + 22y + 4 = 0

Rotation of a Conic In Exercises 4552, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. 8x2 + 8xy + 8y2 + 102x + 262y + 31 = 0

Rotation of a Conic In Exercises 4552, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. xy + x 2y + 3 = 0

Rotation of a Conic In Exercises 4552, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. 5x2 2xy + 5y 2 + 102x = 0

Rotation of a Quadric Surface In Exercises 5356, find the matrix A of the quadratic form associated with the equation. Then find the equation of the quadric surface in the rotated xyz-system. 3x2 2xy + 3y2 + 8z2 16 = 0

Rotation of a Quadric Surface In Exercises 5356, find the matrix A of the quadratic form associated with the equation. Then find the equation of the quadric surface in the rotated xyz-system. 2x2 + 2y2 + 2z2 + 2xy + 2xz + 2yz 1 = 0

Rotation of a Quadric Surface In Exercises 5356, find the matrix A of the quadratic form associated with the equation. Then find the equation of the quadric surface in the rotated xyz-system. x2 + 2y2 + 2z2 + 2yz 1 = 0

Rotation of a Quadric Surface In Exercises 5356, find the matrix A of the quadratic form associated with the equation. Then find the equation of the quadric surface in the rotated xyz-system. x2 + y2 + z2 + 2xy 8 = 0

Constrained Optimization In Exercises 5766, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. z = 3x1 2 + 2x2 2; .x. 2 = 1

Constrained Optimization In Exercises 5766, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. z = 11x1 2 + 4x2 2; .x. 2 = 1

Constrained Optimization In Exercises 5766, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. . z = x1 2 + 12x2 2; 4x1 2+ 25x2 2 = 100

Constrained Optimization In Exercises 5766, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. z = 5x2 + 9y2; x2 + 9y2 = 9

Constrained Optimization In Exercises 5766, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. z = 5x2 + 12xy + 5y2; x2 + y2 = 1

Constrained Optimization In Exercises 5766, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. z = 5x1 2 + 12x1x2; .x. 2 = 1

Constrained Optimization In Exercises 5766, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. z = 6x1x2; .x. 2 = 1

Constrained Optimization In Exercises 5766, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. . z = 9xy; 9x2 + 16y2 = 144

Constrained Optimization In Exercises 5766, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. w = x2 + 3y2 + z2 + 2xy + 2xz + 2yz; x2 + y2 + z2 = 1

Constrained Optimization In Exercises 5766, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. w = 2x2 y2 z2 + 4xy 4xz + 8yz; x2 + y2 + z2 = 1

Let P be a 2 2 orthogonal matrix such that P = 1. Show that there exists a number , 0 < 2, such that P = [ cos sin sin cos ].

CAPSTONE (a) Explain how to model population growth using an age transition matrix and an age distribution vector, and how to find a stable age distribution vector. (b) Explain how to use a matrix equation to solve a system of first-order linear differential equations. (c) Explain how to use the Principal Axes Theorem to perform a rotation of axes for a conic and a quadric surface. (d) Explain how to solve a constrained optimization problem.

Use your schools library, the Internet, or some other reference source to find real-life applications of constrained optimization.

In Exercises 1–6, use the age transition matrix L and age distribution vector \(\mathbf{x}_{1}\) to find the age distribution vectors \(x_{2} \text { and } x_{3}\). Then find a stable age distribution vector. \(L=\left[\begin{array}{cc} 0 & 2 \\ \frac{1}{2} & 0 \end{array}\right], \mathbf{x}_{1}=\left[\begin{array}{l} 10 \\ 10 \end{array}\right]\) Text Transcription: x_1 x_2 and x_3 L=[0 2 1/2 0], x_1=[10 10]

In Exercises 1–6, use the age transition matrix L and age distribution vector \(\mathbf{x}_{1}\) to find the age distribution vectors \(x_{2} \text { and } x_{3}\). Then find a stable age distribution vector. \(L=\left[\begin{array}{cc} 0 & 4 \\ \frac{1}{16} & 0 \end{array}\right], \mathbf{x}_{1}=\left[\begin{array}{l} 160 \\ 160 \end{array}\right]\) Text Transcription: x_1 x_2 and x_3 L=[0 4 1/16 0], x_1=[160 160]

In Exercises 1–6, use the age transition matrix L and age distribution vector \(\mathbf{x}_{1}\) to find the age distribution vectors \(x_{2} \text { and } x_{3}\). Then find a stable age distribution vector. \(L=\left[\begin{array}{lll} 0 & 3 & 4 \\ 1 & 0 & 0 \\ 0 & \frac{1}{2} & 0 \end{array}\right], \mathbf{x}_{1}=\left[\begin{array}{l} 12 \\ 12 \\ 12 \end{array}\right]\) Text Transcription: x_1 x_2 and x_3 L=[0 3 4 1 0 0 0 1/2 0], x_1=[12 12 12]

In Exercises 1–6, use the age transition matrix L and age distribution vector \(\mathbf{x}_{1}\) to find the age distribution vectors \(x_{2} \text { and } x_{3}\). Then find a stable age distribution vector. \(L=\left[\begin{array}{ccc} 0 & 2 & 0 \\ \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{2} & 0 \end{array}\right], \mathbf{x}_{1}=\left[\begin{array}{l} 8 \\ 8 \\ 8 \end{array}\right]\) Text Transcription: x_1 x_2 and x_3 L=[0 2 0 1/2 0 0 0 1/2 0], x_1=[8 8 8]

In Exercises 1–6, use the age transition matrix L and age distribution vector \(\mathbf{x}_{1}\) to find the age distribution vectors \(x_{2} \text { and } x_{3}\). Then find a stable age distribution vector. \(L=\left[\begin{array}{cccc} 0 & 2 & 2 & 0 \\ \frac{1}{4} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 0 \end{array}\right], \mathbf{x}_{1}=\left[\begin{array}{c} 100 \\ 100 \\ 100 \\ 100 \end{array}\right]\) Text Transcription: x_1 x_2 and x_3 L=[0 2 2 0 1/4 0 0 0 0 1 0 0 0 0 1/2 0], x_1=[100 100 100 100]

Finding Age Distribution Vectors In Exercises 16, use the age transition matrix L and age distribution vector x1 to find the age distribution vectors x2 and x3. Then find a stable age distribution vector. L = [ 0 1 2 0 0 0 6 0 1 0 0 4 0 0 1 2 0 0 0 0 0 1 2 0 0 0 0 0 ] , x1 = [ 24 24 24 24 24]

A population has the characteristics below. (a) A total of 75% of the population survives the first year. Of that 75%, 25% survive the second year. The maximum life span is 3 years. (b) The average number of offspring for each member of the population is 2 the first year, 4 the second year, and 2 the third year. The population now consists of 160 members in each of the three age classes. How many members will there be in each age class in 1 year? in 2 years?

A population has the characteristics below. (a) A total of 80% of the population survives the first year. Of that 80%, 25% survive the second year. The maximum life span is 3 years. (b) The average number of offspring for each member of the population is 3 the first year, 6 the second year, and 3 the third year. The population now consists of 120 members in each of the three age classes. How many members will there be in each age class in 1 year? in 2 years?

A population has the characteristics below. (a) A total of 60% of the population survives the first year. Of that 60%, 50% survive the second year. The maximum life span is 3 years. (b) The average number of offspring for each member of the population is 2 the first year, 5 the second year, and 2 the third year. The population now consists of 100 members in each of the three age classes. How many members will there be in each age class in 1 year? in 2 years?

Find the limit (if it exists) of \(A^{n} \mathbf{x}_{1}\) as n approaches infinity, where \(A=\left[\begin{array}{cc} 0 & 2 \\ \frac{1}{2} & 0 \end{array}\right] \quad \text { and } \quad \mathbf{x}_{1}=\left[\begin{array}{l} a \\ a \end{array}\right]\) Text Transcription: A^n x_1 A=[0 2 1/2 0] and x_1=[a a]

In Exercises 11–20, solve the system of first-order linear differential equations. \(\begin{array}{l} y_{1}^{\prime}=2 y_{1} \\ y_{2}^{\prime}=y_{2} \end{array}\) Text Transcription: y_1^prime=2y_1 y_2^prime=y_2

In Exercises 11–20, solve the system of first-order linear differential equations. \(\begin{array}{l} y_{1}^{\prime}=-5 y_{1} \\ y_{2}^{\prime}=4 y_{2} \end{array}\) Text Transcription: y_1^prime=-5y_1 y_2^prime=4y_2

In Exercises 11–20, solve the system of first-order linear differential equations. \(\begin{array}{l} y_{1}^{\prime}=-4 y_{1} \\ y_{2}^{\prime}=-\frac{1}{2} y_{2} \end{array}\) Text Transcription: y_1^prime=-4y_1 y_2^prime=-1/2 y_2

In Exercises 11–20, solve the system of first-order linear differential equations. \(\begin{array}{l} y_{1}^{\prime}=\frac{1}{2} y_{1} \\ y_{2}^{\prime}=\frac{1}{8} y_{2} \end{array}\) Text Transcription: y_1^prime=1/2 y_1 y_2^prime=1/8 y_2

In Exercises 11–20, solve the system of first-order linear differential equations. \(\begin{array}{l} y_{1}^{\prime}=-y_{1} \\ y_{2}^{\prime}=6 y_{2} \\ y_{3}^{\prime}=y_{3} \end{array}\) Text Transcription: y_1^prime=-y_1 y_2^prime=6y_2 y_3^prime=y_3

In Exercises 11–20, solve the system of first-order linear differential equations. \(\begin{array}{l} y_{1}^{\prime}=5 y_{1} \\ y_{2}^{\prime}=-2 y_{2} \\ y_{3}^{\prime}=-3 y_{3} \end{array}\) Text Transcription: y_1^prime=5y_1 y_2^prime=-2y_2 y_3^prime=-3y_3

In Exercises 11–20, solve the system of first-order linear differential equations. \(\begin{array}{l} y_{1}^{\prime}=-0.3 y_{1} \\ y_{2}^{\prime}=0.4 y_{2} \\ y_{3}^{\prime}=-0.6 y_{3} \end{array}\) Text Transcription: y_1^prime=-0.3y_1 y_2^prime=0.4y_2 y_3^prime=-0.6y_3

In Exercises 11–20, solve the system of first-order linear differential equations. \(\begin{array}{l} y_{1}^{\prime}=-\frac{2}{3} y_{1} \\ y_{2}^{\prime}=-\frac{3}{5} y_{2} \\ y_{3}^{\prime}=-8 y_{3} \end{array}\) Text Transcription: y_1^prime=-2/3 y_1 y_2^prime=-3/5 y_2 y_3^prime=-8y_3

In Exercises 11–20, solve the system of first-order linear differential equations. \(\begin{array}{l} y_{1}^{\prime}=7 y_{1} \\ y_{2}^{\prime}=9 y_{2} \\ y_{3}^{\prime}=-7 y_{3} \\ y_{4}^{\prime}=-9 y_{4} \end{array}\) Text Transcription: y_1^prime=7y_1 y_2^prime=9y_2 y_3^prime=-7y_3 y_4^prime=-9y_4

In Exercises 11–20, solve the system of first-order linear differential equations. \(\begin{array}{l} y_{1}^{\prime}=-0.1 y_{1} \\ y_{2}^{\prime}=-\frac{7}{4} y_{2} \\ y_{3}^{\prime}=-2 \pi y_{3} \\ y_{4}^{\prime}=\sqrt{5} y_{4} \end{array}\) Text Transcription: y_1^prime=-0.1 y_1 y_2^prime=-7/4 y_2 y_3^prime=-2 pi y_3 y_4^prime=sqrt 5y_4

In Exercises 21–28, solve the system of first-order linear differential equations. \(\begin{array}{l} y_{1}^{\prime}=y_{1}-4 y_{2} \\ y_{2}^{\prime}=\quad 2 y_{2} \end{array}\) Text Transcription: y_1^prime=y_1-4y_2 y_2^prime= 2y_2

In Exercises 21–28, solve the system of first-order linear differential equations. \(\begin{array}{l} y_{1}^{\prime}=y_{1}-4 y_{2} \\ y_{2}^{\prime}=-2 y_{1}+8 y_{2} \end{array}\) Text Transcription: y_1^prime=y_1-4y_2 y_2^prime=-2y_1+8y_2

In Exercises 21–28, solve the system of first-order linear differential equations. \(\begin{array}{l} y_{1}^{\prime}=y_{1}+2 y_{2} \\ y_{2}^{\prime}=2 y_{1}+y_{2} \end{array}\) Text Transcription: y_1^prime=y_1+2y_2 y_2^prime=2y_1+y_2

In Exercises 21–28, solve the system of first-order linear differential equations. \(\begin{array}{l} y_{1}^{\prime}=y_{1}-y_{2} \\ y_{2}^{\prime}=2 y_{1}+4 y_{2} \end{array}\) Text Transcription: y_1^prime=y_1-y_2 y_2^prime=2y_1+4y_2

In Exercises 21–28, solve the system of first-order linear differential equations. \(\begin{array}{lr}y_1^{\prime}=&y_1-2y_2+y_3\\ y_2^{\prime}=&2y_2+4y_3\\ y_3^{\prime}=&3y_3\end{array}\) Text Transcription: y_1^prime=y_1-2y_2+y_3 y_2^prime=2y_2+4y_3 y_3^prime=3y_3

In Exercises 21–28, solve the system of first-order linear differential equations. \(\begin{array}{l}y_1^{\prime}=2y_1+y_2+y_3\\ y_2^{\prime}=y_1+y_2\\ y_3^{\prime}=y_1\ \ \ \ \ \ \ \ \ +y_3\end{array}\) Text Transcription: y_1^prime=2y_1+y_2+y_3 y_2^prime=y_1+y_2 y_3^prime=y_1 +y_3

In Exercises 21–28, solve the system of first-order linear differential equations. \(\begin{aligned} &y_{1}^{\prime}=\quad 3 y_{2}-5 y_{3}\\ &y_{2}^{\prime}=4 y_{1}-4 y_{2}+10 y_{3}\\ &y_{3}{ }^{\prime}=-4 y_{3} \end{aligned}\) Text Transcription: y_1^prime= 3y_2-5y_3 y_2^prime=4y_1-4y_2+10y_3 y_3^prime=-4y_3

In Exercises 21–28, solve the system of first-order linear differential equations. \(\begin{array}{lr} y_{1}^{\prime}=-2 y_{1} & +y_{3} \\ y_{2}^{\prime}= & 3 y_{2}+4 y_{3} \\ y_{3}^{\prime}= & y_{3} \end{array}\) Text Transcription: y_1^prime=-2y_1 +y_3 y_2^prime= 3y_2+4y_3 y_3^prime= y_3

In Exercises 29–32, write the system of first-order linear differential equations represented by the matrix equation y’ = Ay. Then verify the general solution. \(A=\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right] \begin{array}{l} y_{1}=C_{1} e^{t}+C_{2} t e^{t} \\ y_{2}=C_{2} e^{t} \end{array}\) Text Transcription: A=[1 1 0 1] y_1=C_1e^t+C_2t e^t y_2=C_2 e^t

In Exercises 29–32, write the system of first-order linear differential equations represented by the matrix equation y’ = Ay. Then verify the general solution. \(A=\left[\begin{array}{rr} 1 & -1 \\ 1 & 1 \end{array}\right], \begin{array}{l} y_{1}=C_{1} e^{t} \cos t+C_{2} e^{t} \sin t \\ y_{2}=-C_{2} e^{t} \cos t+C_{1} e^{t} \sin t \end{array}\) Text Transcription: A=[1 -1 1 1], y_1=C_1 e^t cos t+C_2 e^t sin t y_2=-C_2 e^t cos t+C_1 e^t sin t

In Exercises 29–32, write the system of first-order linear differential equations represented by the matrix equation y’ = Ay. Then verify the general solution. \(\begin{array}{l} A=\left[\begin{array}{rrr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & -4 & 0 \end{array}\right], \\ y_{1}=C_{1}+C_{2} \cos 2 t+C_{3} \sin 2 t \\ y_{2}=\quad 2 C_{3} \cos 2 t-2 C_{2} \sin 2 t \\ y_{3}=\quad-4 C_{2} \cos 2 t-4 C_{3} \sin 2 t \end{array}\) Text Transcription: A=[0 1 0 0 0 1 0 -4 0], y_1=C_1+C_2 cos 2t+C_3 sin 2t y_2= 2C_3 cos 2 t-2 C_2 sin 2t y_3=-4C_2 cos 2t-4C_3 sin 2t

In Exercises 29–32, write the system of first-order linear differential equations represented by the matrix equation y’ = Ay. Then verify the general solution. \(\begin{array}{l} A=\left[\begin{array}{rrr} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & -3 & 3 \end{array}\right],\\ y_{1}=C_{1} e^{t}+\quad C_{2} t e^{t}+C_{3} t^{2} e^{t}\\ y_{2}=\quad\left(C_{1}+C_{2}\right) e^{t}+\left(C_{2}+2 C_{3}\right) t e^{t}+C_{3} t^{2} e^{t}\\ y_{3}=\left(C_{1}+2 C_{2}+2 C_{3}\right) e^{t}+\left(C_{2}+4 C_{3}\right) t e^{t}+C_{3} t^{2} e^{t} \end{array}\) Text Transcription: A=[0 1 0 0 0 1 1 -3 3], y_1=C_1e^t+ C_2te^t+C_3t^2e^t y_2=(C_1+C_2)e^t+(C_2+2C_3)te^t+C_3t^2e^t y_3=(C_1+2C_2+2C_3)e^t+(C_2+4C_3)te^t+C_3 t^2 e^t

In Exercises 33–38, find the matrix A of the quadratic form associated with the equation. \(x^{2}+y^{2}-4=0\) Text Transcription: x^2+y^2-4=0

In Exercises 33–38, find the matrix A of the quadratic form associated with the equation. \(x^{2}-4 x y+y^{2}-4=0\) Text Transcription: x^2-4 x y+y^2-4=0

In Exercises 33–38, find the matrix A of the quadratic form associated with the equation. \(9 x^{2}+10 x y-4 y^{2}-36=0\) Text Transcription: 9x^2+10xy-4y^2-36=0

In Exercises 33–38, find the matrix A of the quadratic form associated with the equation. \(12 x^{2}-5 x y-x+2 y-20=0\) Text Transcription: 12x^2-5xy-x+2y-20=0

In Exercises 33–38, find the matrix A of the quadratic form associated with the equation. \(10 x y-10 y^{2}+4 x-48=0\) Text Transcription: 10xy-10y^2+4x-48=0

In Exercises 33–38, find the matrix A of the quadratic form associated with the equation. \(16 x^{2}-4 x y+20 y^{2}-72=0\) Text Transcription: 16x^2-4xy+20y^2-72=0

In Exercises 39–44, find the matrix A of the quadratic form associated with the equation. Then find the eigenvalues of A and an orthogonal matrix P such that \(P^{T} A P\) is diagonal. \(2 x^{2}-3 x y-2 y^{2}+10=0\) Text Transcription: P^T AP 2x^2-3xy-2y^2+10=0

In Exercises 39–44, find the matrix A of the quadratic form associated with the equation. Then find the eigenvalues of A and an orthogonal matrix P such that \(P^{T} A P\) is diagonal. \(5 x^{2}-2 x y+5 y^{2}+10 x-17=0\) Text Transcription: P^T AP 5x^2-2xy+5y^2+10x-17=0

In Exercises 39–44, find the matrix A of the quadratic form associated with the equation. Then find the eigenvalues of A and an orthogonal matrix P such that \(P^{T} A P\) is diagonal. \(13 x^{2}+6 \sqrt{3} x y+7 y^{2}-16=0\) Text Transcription: P^T AP 13x^2+6 sqrt 3xy+7y^2-16=0

In Exercises 39–44, find the matrix A of the quadratic form associated with the equation. Then find the eigenvalues of A and an orthogonal matrix P such that \(P^{T} A P\) is diagonal. \(3 x^{2}-2 \sqrt{3} x y+y^{2}+2 x+2 \sqrt{3} y=0\) Text Transcription: P^T AP 3x^2-2 sqrt 3 xy+y^2+2x+2 sqrt 3 y=0

In Exercises 39–44, find the matrix A of the quadratic form associated with the equation. Then find the eigenvalues of A and an orthogonal matrix P such that \(P^{T} A P\) is diagonal. \(16 x^{2}-24 x y+9 y^{2}-60 x-80 y+100=0\) Text Transcription: P^T AP 16x^2-24xy+9y^2-60x-80y+100=0

In Exercises 39–44, find the matrix A of the quadratic form associated with the equation. Then find the eigenvalues of A and an orthogonal matrix P such that \(P^{T} A P\) is diagonal. \(17 x^{2}+32 x y-7 y^{2}-75=0\) Text Transcription: P^T AP 17x^2+32xy-7y^2-75=0

In Exercises 45–52, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. \(13 x^{2}-8 x y+7 y^{2}-45=0\) Text Transcription: 13x^2-8xy+7y^2-45=0

In Exercises 45–52, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. \(x^{2}+4 x y+y^{2}-9=0\) Text Transcription: x^2+4xy+y^2-9=0

In Exercises 45–52, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. \(2 x^{2}-4 x y+5 y^{2}-36=0\) Text Transcription: 2x^2-4xy+5y^2-36=0

In Exercises 45–52, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. \(7 x^{2}+32 x y-17 y^{2}-50=0\) Text Transcription: 7x^2+32xy-17y^2-50=0

In Exercises 45–52, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. \(2 x^{2}+4 x y+2 y^{2}+6 \sqrt{2} x+2 \sqrt{2} y+4=0\) Text Transcription: 2x^2+4xy+2y^2+6 sqrt 2x+2 sqrt 2 y+4=0

Rotation of a Conic In Exercises 4552, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. 8x2 + 8xy + 8y2 + 102x + 262y + 31 = 0

In Exercises 45–52, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. xy + x ? 2y + 3 = 0

In Exercises 45–52, use the Principal Axes Theorem to perform a rotation of axes to eliminate the xy-term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. \(5 x^{2}-2 x y+5 y^{2}+10 \sqrt{2} x=0\) Text Transcription: 5x^2-2xy+5y^2+10 sqrt 2 x=0

In Exercises 53–56, find the matrix A of the quadratic form associated with the equation. Then find the equation of the quadric surface in the rotated x’ y’ z’ -system. \(3 x^{2}-2 x y+3 y^{2}+8 z^{2}-16=0\) Text Transcription: 3x^2-2xy+3y^2+8z^2-16=0

In Exercises 53–56, find the matrix A of the quadratic form associated with the equation. Then find the equation of the quadric surface in the rotated x’ y’ z’ -system. \(2 x^{2}+2 y^{2}+2 z^{2}+2 x y+2 x z+2 y z-1=0\) Text Transcription: 2x^2+2y^2+2z^2+2xy+2xz+2yz-1=0

In Exercises 53–56, find the matrix A of the quadratic form associated with the equation. Then find the equation of the quadric surface in the rotated x’ y’ z’ -system. \(x^{2}+2 y^{2}+2 z^{2}+2 y z-1=0\) Text Transcription: x^2+2y^2+2z^2+2yz-1=0

In Exercises 53–56, find the matrix A of the quadratic form associated with the equation. Then find the equation of the quadric surface in the rotated x’ y’ z’ -system. \(x^{2}+y^{2}+z^{2}+2 x y-8=0\) Text Transcription: x^2+y^2+z^2+2xy-8=0

In Exercises 57–66, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. \(z=3 x_{1}^{2}+2 x_{2}^{2} ;\|\mathbf{x}\|^{2}=1\) Text Transcription: z=3x_1^2+2 x_2^2 ;|x|^2=1

In Exercises 57–66, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. \(z=11 x_{1}^{2}+4 x_{2}^{2} ;\|\mathbf{x}\|^{2}=1\) Text Transcription: z=11x_1^2+4x_2^2 ;|x|^2=1

In Exercises 57–66, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. \(z=x_{1}^{2}+12 x_{2}^{2} ; 4 x_{1}^{2}+25 x_{2}^{2}=100\) Text Transcription: z=x_1^2+12x_2^2 ; 4x_1^2+25x_2^2=100

In Exercises 57–66, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. \(z=-5 x^{2}+9 y^{2} ; x^{2}+9 y^{2}=9\) Text Transcription: z=-5x^2+9y^2 ; x^2+9y^2=9

In Exercises 57–66, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. \(z=5 x^{2}+12 x y+5 y^{2} ; x^{2}+y^{2}=1\) Text Transcription: z=5x^2+12xy+5y^2 ; x^2+y^2=1

In Exercises 57–66, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. \(z=5 x_{1}^{2}+12 x_{1} x_{2} ;\|\mathbf{x}\|^{2}=1\) Text Transcription: z=5x_1^2+12x_1x_2 ;|x|^2=1

In Exercises 57–66, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. \(z=6 x_{1} x_{2} ;\|\mathbf{x}\|^{2}=1\) Text Transcription: z=6x_1x_2 ;|x|^2=1

In Exercises 57–66, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. \(z=9 x y ; 9 x^{2}+16 y^{2}=144\) Text Transcription: z=9xy ; 9x^2+16y^2=144

In Exercises 57–66, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. \(w=x^{2}+3 y^{2}+z^{2}+2 x y+2 x z+2 y z ; x^{2}+y^{2}+z^{2}=1\) Text Transcription: w=x^2+3y^2+z^2+2xy+2xz+2yz ; x^2+y^2+z^2=1

In Exercises 57–66, find the maximum and minimum values, and a vector where each occurs, of the quadratic form subject to the constraint. \(w=2 x^{2}-y^{2}-z^{2}+4 x y-4 x z+8 y z ; x^{2}+y^{2}+z^{2}=1\) Text Transcription: w=2x^2-y^2-z^2+4xy-4xz+8yz ; x^2+y^2+z^2=1

Let P be a 2 2 orthogonal matrix such that P = 1. Show that there exists a number , 0 < 2, such that P = [ cos sin sin cos ].

(a) Explain how to model population growth using an age transition matrix and an age distribution vector, and how to find a stable age distribution vector. (b) Explain how to use a matrix equation to solve a system of first-order linear differential equations. (c) Explain how to use the Principal Axes Theorem to perform a rotation of axes for a conic and a quadric surface. (d) Explain how to solve a constrained optimization problem.

Use your school’s library, the Internet, or some other reference source to find real-life applications of constrained optimization.