Question

Notify Me Bookmark

Match each initial value problem with the phase plane plot of its solution. (The arrows on the curves indicate how the solution point moves as t increases.)? 1. y→′ = [0 2 −2 0]y→, y→(0) = [1 0]. 2. y→′ = [0 −1 1 0]y→, y→(0) = [1 0]. 3. y→′ = [−1 −0.5 0.5 −1]y→, y→(0) = [1 0]. 4. y→′ = [1 −0.5 0.5 1]y→, y→(0) = [1 0].

Match each initial value problem with the phase plane plot of its solution. (The arrows on the curves indicate how the solution point moves as t increases.)? 1. y→′ = [0 2 −2 0]y→, y→(0) = [1 0]. 2. y→′ = [0 −1 1 0]y→, y→(0) = [1 0]. 3. y→′ = [−1 −0.5 0.5 −1]y→, y→(0) = [1 0]. 4. y→′ = [1 −0.5 0.5 1]y→, y→(0) = [1 0].

Image text
Match each initial value problem with the phase plane plot of its solution. (The arrows on the curves indicate how the solution point moves as t increases.) ? ∨ 1 . y → ′ = [ 0 2 − 2 0 ] y → , y → ( 0 ) = [ 1 0 ] . ◻ 2. y → ′ = [ 0 − 1 1 0 ] y → , y → ( 0 ) = [ 1 0 ] . ◻ 3. y → ′ = [ − 1 − 0.5 0.5 − 1 ] y → , y → ( 0 ) = [ 1 0 ] . ◻ 4. y → ′ = [ 1 − 0.5 0.5 1 ] y → , y → ( 0 ) = [ 1 0 ] . A C B

Detailed Answer

0 0

Please enter your email here. You will get email when this question is answered by our tutor.

Doubtrix Logo

Problem is not yet solved!

Click on Notify me to get email when question is answered.

Join Doubtrix with 7-days free trial

  • Icon Correct & Verified Answers
  • Icon No AI-Generated Answers
  • Icon Video Solution for Difficult Questions
  • Icon Work With Experts to Reach at Correct Answers
Notify me Login Sign Up

Similar/Related Questions

5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]5. (7 points) In this problem we'll complete the first few steps of solving a differential equation in context using eigenvectors and eigenvalues. Suppose a car rental company has two locations, location P and location Q. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. Market research determines that 80% of the cars rented at location P are returned to P and 20% are returned to Q. 40% of the cars rented at location Q are returned to Q and 60% are returned to P. (a) Suppose that there are 1000 cars at location P and no cars at location Q on Monday morning. How many cars are there at location P and Q respectively at the end of the day on Monday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (b) How many cars are at each location, P and Q, at the end of the day on Tuesday? (c) If we let Pk and Qk be the number of cars at locations P and Q, respectively, at the end of day k, then we have Pk+1 = 0.8Pk + 0.6Qk Qk+1 = 0.2Pk + 0.4Qk Explain why this system of equations appropriately models the given situation. (d) Write the above system of equations as a matrix equation. (e) Suppose that v→ = [3 1] and w→ = [−1 1]. Compute Av→ and Aw→ to show that v→ and w→ are eigenvectors of A (where A is the coefficient matrix you found in part (d)). (f) What are the associated eigenvalues? Let λ1 be the eigenvalue associated with v→ and λ2 be the eigenvalue associated with w→. (g) We said that 1000 cars are initially at location P and none at location Q. This means that the initial vector describing the number of cars is [P0 Q0] = [1000 0] Write this initial condition as a linear combination of v→ and w→. To do this, consider [P0 Q0] = c1v→ + c2w→ and solve for c1 and c2. Then, substitute those values back into this matrix equation. [P0 Q0] = …−−−−−−[3 1]+…−−−−−−−[−1 1]
Solution
0 0

4. (6 points) For each of the following systems, classify the equilibrium type of the origin, indicate whether it is stable and not asymptotically stable, unstable, or asymptotically stable, and draw the phase plane. (a) y→′ = [1 2 3 −4]y→ (b) y→′ = [2 2 −2 2]y→4. (6 points) For each of the following systems, classify the equilibrium type of the origin, indicate whether it is stable and not asymptotically stable, unstable, or asymptotically stable, and draw the phase plane. (a) y→′ = [1 2 3 −4]y→ (b) y→′ = [2 2 −2 2]y→
Solution
0 0

Suppose that the matrix A has the following eigenvalues and eigenvectors: λ1 = 1 with v→1 = [1 0]. and λ2 = −1 with v→2 = [4 1]. Write the solution to the linear system r→′ = Ar→ in the following forms. A. In eigenvalue/eigenvector form: [x(t) y(t)] = c1[ ]et + c2[ ]et B. In fundamental matrix form: [x(t)y(t)] = [ ][c1 c2] C. As two equations: (write "c1" and "c2" for c1 and c2 ) x(t) = y(t) = Note: if you are feeling adventurous you could use other eigenvectors like 4v→1 or −3v→2.Suppose that the matrix A has the following eigenvalues and eigenvectors: λ1 = 1 with v→1 = [1 0]. and λ2 = −1 with v→2 = [4 1]. Write the solution to the linear system r→′ = Ar→ in the following forms. A. In eigenvalue/eigenvector form: [x(t) y(t)] = c1[ ]et + c2[ ]et B. In fundamental matrix form: [x(t)y(t)] = [ ][c1 c2] C. As two equations: (write "c1" and "c2" for c1 and c2 ) x(t) = y(t) = Note: if you are feeling adventurous you could use other eigenvectors like 4v→1 or −3v→2.
Solution
0 0