Answer: In each of 1 through 5, (a) verify that the given functions satisfy the system

In each of Problems 1 through 5, (a) verify that the given functions satisfy the system, (b) write the system in matrix form X= AX for an appropriate A, (c) write n linearly independent n 1 matrix solutions 1, , n , for appropriate n, (d) use the determinant test of Theorem 10.2(2) to verify that these solutions are linearly independent, (e) form a fundamental matrix for the system, and (f) use the fundamental matrix to solve the initial value problemx 1 = 5x1 + 3x2, x 2 = x1 + 3x2, x1(t)= c1e2t + 3c2e6t , x2(t) = c1e2t + c2e6t , x1(0)= 0, x2(0) = 4

In each of Problems 1 through 5, (a) verify that the given functions satisfy the system, (b) write the system in matrix form X= AX for an appropriate A, (c) write n linearly independent n 1 matrix solutions 1, , n , for appropriate n, (d) use the determinant test of Theorem 10.2(2) to verify that these solutions are linearly independent, (e) form a fundamental matrix for the system, and (f) use the fundamental matrix to solve the initial value problemx 1 = 2x1 + x2, x 2 = 3x1 + 6x2, x1(t)= c1e4t cos(t) + c2e4t sin(t) x2(t)= 2c1e4t [cos(t) sin(t)] +2c2e4t [cos(t) + sin(t)], x1(0)= 2, x2(0) = 1

In each of Problems 1 through 5, (a) verify that the given functions satisfy the system, (b) write the system in matrix form X= AX for an appropriate A, (c) write n linearly independent n 1 matrix solutions 1, , n , for appropriate n, (d) use the determinant test of Theorem 10.2(2) to verify that these solutions are linearly independent, (e) form a fundamental matrix for the system, and (f) use the fundamental matrix to solve the initial value problemx 1 = 3x1 + 8x2, x 2 = x1 x2, x1(t)= 4c1e(1+2 3)t + 4c2e(12 3t) , x2(t)= (1 + 3)c1e(1+ 3)t + (1 3)c2e(12 3)6t , x1(0)= 2, x2(0) = 2

In each of Problems 1 through 5, (a) verify that the given functions satisfy the system, (b) write the system in matrix form X= AX for an appropriate A, (c) write n linearly independent n 1 matrix solutions 1, , n , for appropriate n, (d) use the determinant test of Theorem 10.2(2) to verify that these solutions are linearly independent, (e) form a fundamental matrix for the system, and (f) use the fundamental matrix to solve the initial value problemx 1 = x1 x2, x 2 = 4x1 + 2x2, x1(t)= 2e3t/2 c1 cos( 15t/2) + c2 sin( 15t/2) , x2(t)= c1e3t/2 cos( 15t/2) + 15 sin( 15t/2) c2e3t/2 sin( 15t/2) + 15 cos( 15t/2) , x1(0)= 2, x2(0) = 7

In each of Problems 1 through 5, (a) verify that the given functions satisfy the system, (b) write the system in matrix form X= AX for an appropriate A, (c) write n linearly independent n 1 matrix solutions 1, , n , for appropriate n, (d) use the determinant test of Theorem 10.2(2) to verify that these solutions are linearly independent, (e) form a fundamental matrix for the system, and (f) use the fundamental matrix to solve the initial value problemx 1 = 5x1 4x2 + 4x3, x 2 = 12x1 11x2 + 12x3, x 3(t)= 4x1 4x2 + 5x3 x1(t)= c1et + c3e3t , x2(t) = c2e2t + c3e3t , x3(t)= (c3 c1)et + c3e3t , x1(0)= 1, x2(0) = 3, x3(0) = 5 1

In each of Problems 1 through 10, find a fundamental matrix for the system and write the general solution as a matrix. If initial values are given, solve the initial value problemx 1 = 3x1, x 2 = 5x1 4x2

In each of Problems 1 through 10, find a fundamental matrix for the system and write the general solution as a matrix. If initial values are given, solve the initial value problemx 1 = 4x1 + 2x2, x 2 = 3x1 + 3x2

In each of Problems 1 through 10, find a fundamental matrix for the system and write the general solution as a matrix. If initial values are given, solve the initial value problemx 1 = x1 + x2, x 2 = x1 + x2

In each of Problems 1 through 10, find a fundamental matrix for the system and write the general solution as a matrix. If initial values are given, solve the initial value problemx 1 = 2x1 + x2 2x3, x 2 = 3x1 2x2, x 3 = 3x1 x2 3x3 5

In each of Problems 1 through 10, find a fundamental matrix for the system and write the general solution as a matrix. If initial values are given, solve the initial value problem. x 1 = x1 + 2x2 + x3, x 2 = 6x1 x2, x 3 = x1 2x2 x3

In each of Problems 1 through 10, find a fundamental matrix for the system and write the general solution as a matrix. If initial values are given, solve the initial value problemx 1 = 3x1 4x2, x 2 = 2x1 3x2; x1(0) = 7, x2(0) = 5

In each of Problems 1 through 10, find a fundamental matrix for the system and write the general solution as a matrix. If initial values are given, solve the initial value problemx 1 = x1 2x2, x 2 = 6x1; x1(0) = 1, x2(0) = 19

In each of Problems 1 through 10, find a fundamental matrix for the system and write the general solution as a matrix. If initial values are given, solve the initial value problemx 1 = 2x1 10x2, x 2 = x1 x2; x1(0) = 3, x2(0) = 6

In each of Problems 1 through 10, find a fundamental matrix for the system and write the general solution as a matrix. If initial values are given, solve the initial value problemx 1 = 3x1 x2 + x3, x 2 = x1 + x2 x3, x 3 = x1 x2 + x3; x1(0) = 1, x2(0) = 5, x3(0) = 1 1

In each of Problems 1 through 10, find a fundamental matrix for the system and write the general solution as a matrix. If initial values are given, solve the initial value problemx 1 = 2x1 + x2 x3, x 2 = 3x1 2x2, x 3 = 3x1 + x2 3x3; x1(0) = 1, x2(0) = 7, x3(0) = 3 I

In each of Problems 11 through 15, find a real-valued fundamental matrix for the system X= AX with the given coefficient matrix.2 4 1 2

In each of Problems 11 through 15, find a real-valued fundamental matrix for the system X= AX with the given coefficient matrix.0 5 1 2

In each of Problems 11 through 15, find a real-valued fundamental matrix for the system X= AX with the given coefficient matrix.3 5 1 1

In each of Problems 11 through 15, find a real-valued fundamental matrix for the system X= AX with the given coefficient matrix. 1 1 1 1 1 0 1 0 1

In each of Problems 11 through 15, find a real-valued fundamental matrix for the system X= AX with the given coefficient matrix. 21 0 50 0 0 3 2

In each of Problems 16 through 21, find a fundamental matrix for the system with the given coefficient matrix.2 0 5 2

In each of Problems 16 through 21, find a fundamental matrix for the system with the given coefficient matrix.3 2 0 3

In each of Problems 16 through 21, find a fundamental matrix for the system with the given coefficient matrix. 150 010 481

In each of Problems 16 through 21, find a fundamental matrix for the system with the given coefficient matrix. 25 6 08 9 0 1 2

In each of Problems 16 through 21, find a fundamental matrix for the system with the given coefficient matrix. 0 1 00 0 0 10 0 0 01 1 200

In each of Problems 16 through 21, find a fundamental matrix for the system with the given coefficient matrix. 1 5 2 6 03 0 4 03 0 4 00 0 1

In each of Problems 1 through 9, use variation of parameters to find the general solution, with A and G given. If initial conditions are given, also satisfy the initial value problem5 2 2 1 ,  3et e3t

In each of Problems 1 through 9, use variation of parameters to find the general solution, with A and G given. If initial conditions are given, also satisfy the initial value problem4 1 2 , 1 3t

In each of Problems 1 through 9, use variation of parameters to find the general solution, with A and G given. If initial conditions are given, also satisfy the initial value problem7 1 1 5 , 2e6t 6te6t

In each of Problems 1 through 9, use variation of parameters to find the general solution, with A and G given. If initial conditions are given, also satisfy the initial value problem 20 0 0 6 4 0 4 2 , e2t cos(3t) 2 2

In each of Problems 1 through 9, use variation of parameters to find the general solution, with A and G given. If initial conditions are given, also satisfy the initial value problem 1 000 4 300 0 030 1291 , 0 2et 0 et

In each of Problems 1 through 9, use variation of parameters to find the general solution, with A and G given. If initial conditions are given, also satisfy the initial value problem2 0 5 2 , 2 10t ;  0 3

In each of Problems 1 through 9, use variation of parameters to find the general solution, with A and G given. If initial conditions are given, also satisfy the initial value problem5 4 4 3 ,  2et 2et ;  1 3

In each of Problems 1 through 9, use variation of parameters to find the general solution, with A and G given. If initial conditions are given, also satisfy the initial value problem 2 3 1 024 001 , 10e2t 6e2t e2t ; 5 11 2

In each of Problems 1 through 9, use variation of parameters to find the general solution, with A and G given. If initial conditions are given, also satisfy the initial value problem 1 3 0 3 5 0 4 7 2 , te2t te2t t 2 e2t ; 6 2 3

In each of Problems 10 through 19, find a general solution of the system. If initial values are given, also solve the initial value problemx 1 = 2x1 + x2, x 2 = 4x1 + 3x2 + 10 cos(t)

In each of Problems 10 through 19, find a general solution of the system. If initial values are given, also solve the initial value problemx 1 = 3x1 + 3x2 + 8, x 2 = x1 + 5x2 + 4e3t

In each of Problems 10 through 19, find a general solution of the system. If initial values are given, also solve the initial value problemx 1 = x1 + x2 + 6e3t , x 2 = x1 + x2 + 4

In each of Problems 10 through 19, find a general solution of the system. If initial values are given, also solve the initial value problemx 1 = 6x1 + 5x2 4 cos(3t), x 2 = x1 + 2x2 + 8

In each of Problems 10 through 19, find a general solution of the system. If initial values are given, also solve the initial value problemx 1 = 3x1 2x2 + 3e2t , x 2 = 9x1 3x2 + e2t

In each of Problems 10 through 19, find a general solution of the system. If initial values are given, also solve the initial value problemx 1 = x1 + x2 + 6e2t , x 2 = x1 + x2 + 2e2t ; x1(0) = 6, x2(0) = 0

In each of Problems 10 through 19, find a general solution of the system. If initial values are given, also solve the initial value problemx 1 = x1 2x2 + 2t, x 2 = x1 + 2x2 + 5; x1(0) = 13, x2(0) = 12

In each of Problems 10 through 19, find a general solution of the system. If initial values are given, also solve the initial value problemx 1 = 2x1 5x2 + 5 sin(t), x 2 = x1 2x2; x1(0) = 10, x2(0) = 5

In each of Problems 10 through 19, find a general solution of the system. If initial values are given, also solve the initial value problemx 1 = 5x1 4x2 + 4x3 3e3t , x 2 = 12x1 11x2 + 12x3 + t, x 3 = 4x1 4x2 + 5x3; x1(0) = 1, x2(0) = 1, x3(0) = 2 1

In each of Problems 10 through 19, find a general solution of the system. If initial values are given, also solve the initial value problemx 1 = 3x1 x2 x3, x 2 = x1 + x2 x3 + t, x 3 = x1 x2 + x3 + 2et ; x1(0) = 1, x2(0) = 2, x3(0) = 2 C

A =  1 1 5 1

A =  2 1 2 1

A =  5 2 4 8

A =  4 1 2 2

A = 1 01 211 1 1 0

Let D be an n n diagonal matrix of numbers, with jth diagonal element dj . Show that eDt is the n n diagonal matrix having ed j t as its jth diagonal element.

Let A be an n n matrix of numbers, and let P be an n n nonsingular matrix of numbers. Let B=P1 AP. Show that eBt = P1 eAt P. From this, conclude that eAt = PeBt P1 .

Use the results of Problems 6 and 7 to show that, if P diagonalizes A, so P1 AP = D, which is a diagonal matrix with diagonal elements dj . Then eAt = PeDt P1 , where eDt is the diagonal matrix having ed j t as main diagonal elements.

Use the result of Problem 8 to determine the exponential matrix in each of Problems 1 and 2

Referring to the circuit of Figure 10.4, determine how much time elapses between the time the switch is closed and the time the charge on the capacitor is a maximum. What is the maximum voltage on the capacitor?

Referring to Figure 10.5, tank 1 initially contains 200 gallons of saltwater (brine), while tank 2 initially contains 300 gallons of brine. Beginning at time 0, brine is pumped into tank 1 at the rate of 4 gallons per minute, pure water is pumped into tank 2 at 6 gallons per minute, and the brine solutions are interchanged between the two tanks and also flow out of both tanks at the rates shown. The input to tank 1 contains 1/4 pound of salt per gallon, tank 1 initially has 200 pounds of salt, and tank 2 initially has 150 pounds of salt. Determine the amount of salt in each tank at time t > 0.

Two tanks are connected as shown in Figure 10.6. Tank 1 initially contains 100 gallons of water in which 40 pounds of salt are dissolved. Tank 2 initially contains 150 gallons of pure water. Beginning at t = 0, a brine solution containing 1/5 pound of salt per gallon is pumped into tank 1 at the rate of 5 gallons per minute. At this time, a solution which also contains 1/5 pound of salt per gallon is pumped into tank 2 at the rate of 10 gallons per minute. The brine solutions are interchanged between the tanks and also flow out of both tanks at the rates shown. Determine the amount of salt in each tank for t 0. Also calculate the time at which the brine solution in tank 1 reaches its minimum salinity (concentration of salt) and determine how much salt is in tank 1 at that time.

Find the currents i1(t) and i2(t) in the circuit of Figure 10.7 for t > 0, assuming that the currents and charges are all zero prior to the switch being closed at t = 0. 50 1 H 103 F i2 i1 5 V FIGURE 10.7 Circuit for Problem 4, Section 10.5. Each of Problems 5 and 6 refer to the system of Figure 10.8. Derive and solve the differential equations for the motions of the masses under the assumption that there is no damping

Each mass is pulled downward one unit and released from rest with no external driving forces.

The masses have zero initial displacement and velocity. The lower mass is subjected to an external driving force of magnitude F(t) = 2 sin(3t), while the upper mass has no driving force applied to it.

Refer to the mechanical system of Figure 10.9. The left mass is pushed to the right one unit, and the right mass is pushed to the left one unit. Both are released from rest at time t = 0. Assume that there are no external driving forces. Derive and solve the differential equations with appropriate initial conditions for the displacement of the masses, assuming that there is no damping. Denote left to right as the positive direction.

Find the currents in each loop of the circuit of Figure 10.10. Assume that the currents and charges are all zero prior to the switch being closed at t = 0.

In the circuit of Figure 10.11, assume that the currents and charges are all zero prior to the switch being closed at time 0. Find the loop currents for time t > 0. 50 103 F 10 H i2

Find the loop currents in the circuit of Figure 10.12 for t >0, assuming that the currents and charge are all zero prior to the switch being closed at t = 0. Also determine the maximum value of Eout(t) and when this maximum value is reached. 20 25 25 1/50 F 10 H Eout i1 45 V i2 i3

Derive a system of differential equations for the displacement functions for the masses in Figure 10.13, in which a =1026. Assume that the top weight is lowered one unit and the lower one raised one unit, then both are released from rest at time 0. The upper weight is free of external driving forces, while the lower weight is subjected to an external force of magnitude F(t) = 39 sin(t). k1 = 65 a y1 y2 k2 = a k3 = 65 a m1 = 5 m2 = 13 FIGURE 10.13 Mass/

In each of Problems 1 through 10, classify the origin of the system X= AX for the given coefficient matrix. If software is available, produce a phase portrait.A =  3 5 5 7

In each of Problems 1 through 10, classify the origin of the system X= AX for the given coefficient matrix. If software is available, produce a phase portrait.A =  1 4 3 0

In each of Problems 1 through 10, classify the origin of the system X= AX for the given coefficient matrix. If software is available, produce a phase portrait.A =  1 5 1 1

In each of Problems 1 through 10, classify the origin of the system X= AX for the given coefficient matrix. If software is available, produce a phase portrait.A =  9 7 6 4

In each of Problems 1 through 10, classify the origin of the system X= AX for the given coefficient matrix. If software is available, produce a phase portrait.A =  7 17 2 1

In each of Problems 1 through 10, classify the origin of the system X= AX for the given coefficient matrix. If software is available, produce a phase portrait.A =  2 7 5 10

In each of Problems 1 through 10, classify the origin of the system X= AX for the given coefficient matrix. If software is available, produce a phase portrait.A =  4 1 1 2

In each of Problems 1 through 10, classify the origin of the system X= AX for the given coefficient matrix. If software is available, produce a phase portrait.A =  3 5 8 3

In each of Problems 1 through 10, classify the origin of the system X= AX for the given coefficient matrix. If software is available, produce a phase portrait.A =  2 1 3 2

In each of Problems 1 through 10, classify the origin of the system X= AX for the given coefficient matrix. If software is available, produce a phase portrait.A =  6 7 7 20

Derive a system of differential equations modeling the predator/prey relationship in an environment with indiscriminate harvesting. Do this by assuming that there is some outside agent that removes numbers of both species from the system at a rate proportional to the populations, with the same constant of proportionality for both specie

Use a software package to generate a phase portrait for each of the following predator/prey models. (a) x= x 0.5x y, y= 2x y 1.2y (b) x= 3x 1.5x y, y= x y 1.6y (c) x= 1.6x 2.1x y, y= 1.9x y 0.4y (d) x= 1.8 0.2x y, y= 3.1x y 0.4y 13. Ge

Generate a phase portrait for each of the following competing species models. (a) x= 2x x y, y= y 2x y (b) x= 1.6y 1.2x y, y= 2y 0.4x y (c) x= 1.4x 0.6x y, y= 2y 0.7x y (d) x= 3.2x 1.4x y, y= 4.4y 0.8x y 14. A

A more sophisticated approach to a competing species model is to incorporate a logistic term, leading to the model x = ax bx 2 kxy, y = cy dy2 rxy, with the coefficients positive constants. Generate phase portraits for the following systems. (a) x= x(1 x 0.5y), y= y(1 0.5y 0.25x) (b) x= x(1 x 0.2y), y= y(1 0.4y 0.25x) (c) x= x(2 x 0.2y), y= y(1 0.4y x) (d) x= x(1 0.5x y), y= y(2 0.5y 0.4x) Copyrigh