Consider the system of differential equations y1′ = 28y1 − 14y2, y2′ = 35y1 − 14y2. a. Rewrite this system as a matrix equation y→′ = Ay→. y→′ = [ ]y→ b. Compute the eigenvalues of the coefficient matrix A and enter them as a comma separated list.
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.
For the linear system y→′ = [5 6 −3 −4]y→ Find the eigenvalues and eigenvectors for the coefficient matrix. λ1 = , v→1 = [ ], and λ2 = , v→2 = [ ]
Consider the linear system y→′ = [3 2 −5 −3]y→. Find the eigenvalues and eigenvectors for the coefficient matrix. λ1 = , v→1 = [ ], and λ2 = , v→2 = [ ]
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. Find the most general real-valued solution to the linear system of differential equations x→′ = [5 −6 9 −10]x→. [x1(t) x2(t)] = c1[ ] + c2[ ] b. In the phase plane, this system is best described as a source / unstable node sink / stable node saddle center point / ellipses spiral source spiral sink none of these
Suppose y1′ = t5y1 + 4y2 + sec(t), y2′ = sin(t)y1 + ty2 − 2. This system of linear differential equations can be put in the form y′ = P(t)y + g(t). Determine P(t) and g(t). P(t) = [ ] g(t) = [ ]
Consider the linear system y→′ = [−3 −2 5 3]y→. a. Find the eigenvalues and eigenvectors for the coefficient matrix. λ1 = , v→1 = [ ], and λ2 = , v→2 = [ ] b. Find the real-valued solution to the initial value problem {y1′ = −3y1 − 2y2, y1(0) = −2 y2′ = 5y1 + 3y2, y2(0) = 5 Use t as the independent variable in your answers. y1(t) = y2(t) =
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]
1. (4 points) Consider the system of linear equations given below: x1 − x2 + x3 = 8 2x1 + 3x2 − x3 = −2 3x1 − 2x2 − 9x3 = 9 x1 + 3x3 = 7 (a) Write the system in matrix equation form. (b) Solve the system (find the values of x1, x2, x3 ) using reduced row-echelon form (you may use an RREF solver!).
2. (4 points) Consider the matrix given below: [2 4 0 −1 −2 2 1 2 k] (a) Provide a value of k that will result in the matrix having a rank equal to 1, or explain why this is not possible. (b) Provide a value of k that will result in the matrix having a rank equal to 2, or explain why this is not possible.
3. (3 points) Circle the systems below that represent linear systems of differential equations. y1′ = y2 + t x′ = 3 − xy x′ = t2z + t2x y2′ = y3 + t y′ = 4 − yz y′ = t2y + t2z y3′ = y1 + t2 z′ = 5 − xz z′ = t2x + t2y
4. (2 points) Rewrite the following differential equation as a system of first-order linear ODEs: y′′′ + 2y′′ − 3ty′ + 4 = t7
Do the columns of the matrix span R2? Select an Answer 1. A = [2 −4 −5 10] Select an Answer 2. A = [7 −55 4 −5 40 −3] Select an Answer 3. A = [−7 −6 −2 9] Select an Answer 4. A = [−8 32 128 7 −28 −112]
Let H = span{u, v}. For each of the following sets of vectors determine whether H is a line or a plane. Select an Answer 1. u = [−1 −2 2], v = [−4 −8 8] Select an Answer 2. u = [1 −2 1], v = [−4 5 −1] Select an Answer 3. u = [−8 −8 −8], v = [0 0 0] Select an Answer u = [5 5 −4], v = [21 20 −15]
The matrix A = [7 6 −4 −4] has an eigenvalue λ = −1. Find an eigenvector for this eigenvalue. [ ] Note: You should solve the following problem WITHOUT computing all eigenvalues. The matrix B = [−3 −5 −2 −6] has an eigenvector v→ = [−1 0 4]. Find the eigenvalue for this eigenvector. λ =
For the linear system y→′ = [10 12 −6 −8]y→ Find the eigenvalues and eigenvectors for the coefficient matrix. λ1 = , v→1 = [ ], and λ2 = , v→2 = [ ]
Consider the linear system y→′ = [−12 −8 20 12]y→. Find the eigenvalues and eigenvectors for the coefficient matrix. λ1 = , v→1 = [ ], and λ2 = , v→2 = [ ]
Let A = [−4 −5 3], B = [24 45 −33], and C = [8 13 −9] Are A, B and C linearly dependent, or are they linearly independent? Linearly independent Linearly dependent If they are linearly dependent, determine a non-trivial linear relation. Otherwise, if the vectors are linearly independent, enter 0's for the coefficients, since that relationship always holds. A + B + C = 0.