If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane (see the figure) is given by the formula t = 2a gsinθcosθ, where a is the length (in feet) of the base and g≈32 feet per second per second is the acceleration due to gravity. How long does it take a block to slide down an inclined plane with base a = 90 feet when θ = 30∘? t = (Do not round until the final answer. Then round to two decimal places as needed.)
1. ( 6 points) Answer the following questions, and provide a justification for your response. You may provide a specific example, but your justification must address why your answer is true beyond just that example. (a) If every entry in one row of a matrix is 0, what can we say about the determinant value of that matrix? (b) If two rows of a square matrix are identical, what can we say about the determinant value of that matrix? (c) If two columns of a square matrix are identical, what can you say about the determinant value of that matrix?
2. (3 points) Suppose that a 2×2 matrix A has an eigenvalue λ = 1 + 3i with eigenvector [2 + i 1]. Compute the general solution to y→′ = Ay→.
3. (5 points) Let x and y represent the population sizes of two species at a given time. (a) Consider the two systems below, (A) and (B). Which system of equations describes a situation where the two species competitive species (where both species are harmed by interaction) and which system respresents cooperative species (where both species benefit from interaction)? Explain your reasoning. (A) (B) dxdt = −5x + 2xy dxdt = 3x(1 − x3) − 110 xy dydt = −4y + 3xy dydt = 2y(1 − y10) − 15 xy (b) Describe a real-world context in which we might see a system with a nodal sink. Explain why your context fits the conditions for having an equilibrium which is a nodal sink.
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→
1. (2 points each) Answer the following questions: (a) Using the method of undetermined coefficients, what is the most accurate guess for the particular solution to y′′ − 4y′ + 4y = e2t? (b) Let ω∈R be a fixed positive constant. Construct a second-order differential equation whose generalized solution function is: y = c1 cos(ωt) + c2 sin(ωt) (c) Rewrite the following differential equation as a system of first-order linear ODEs: y′′ + 2y′′ − 3ty′ + 4y = 0 (d) Without using Partial Fraction decomposition find the inverse Laplace transform of F(s) = 6s−5 (s2+9) (e) Is the differential equation (3x2y + 2)dx − (x3 + y)dy = 0 exact? Why or why not?
2. Consider an undamped oscillator which moves a certain mass. The movement of the mass can be described by the equation below, where δ(t) represents the Dirac Delta function. y′′ − 9y = δ(t) with y(0) = 0 and y′(0) = 0. (a) ( 8 points) Find the solution function, y(t), for this differential equation using Laplace transforms. (b) (2 points) In the case given by the problem, the Dirac Delta function applies an instantaneous force (impulse) at time t = 0. Write a differential equation to represent the situation if the impulse were instead applied at time t = π.
3. (2 points) Use the below slope field to make a phase line that represents this slope field. (Hint: what are the equilibria? What is the behavior of y′ on either side of each?). Then, classify each equilibria as stable, unstable or semi-stable.
(8 points) Use the below system to answer the following questions. x′ = 2x(1 − x2) − xy y′ = 3y(1 − y3) − 2xy (a) Identify the equations of the Nullclines of the system. (b) Sketch an accurate graph of the Nullclinces of the system. (c) On your sketch label the coordinate points of each equilibrium points. Meaning, identify the x and y values of each equilibrium point.
Use your knowledge of eigenvalues and eigenvectors to answer the following questions. (a) (5 points) Find the eigenvalues and corresponding eigenvectors of the following matrix. (2 7 −1 −6) (b) (5 points) Consider the system y′ = (−15 −18 9 12)y It has eigenvalues/vectors of λ1 = −6, v1 = (−2 1) and λ2 = 3, v2 = (1 −1). What is the general solution for this system?
Use your knowledge of equilibria of systems to answer the following questions. (a) (2 points) Consider the system x′ = (−1 2 −5 0)x Classify the equilibrium point x = (0, 0) as a node/spiral, sink/source, saddle point or center. (b) (2 points) Consider the system z′ = (3 1 −1 3)z Classify the equilibrium point z = (0, 0) as a node/spiral, sink/source, saddle point or center.
Let x ≥ 0, y ≥ 0, r > 1 and consider the non-linear system x′ = −x + rxy − x2 y′ = y(1 + y) (a) (2 points) Compute the Jacobian of this system. b) (4 points) The point (r−1, 1) is an equilibrium point of this system. Use supportive work to identify (r−1, 1) as sink/source, nodal/spiral, saddle point or center.