Consider the initial value problem dy 3 2- y = 3t + 2e', y(0) = yo...
2y (9 points) Given the initial value problem y' => y (xo) = yo. Use the existence and uniqueness theorem to show that a) a unique solution exists on any interval where xo + 0, b) no solution exists if y (0) = yo # 0, and c) an infinite number of solutions exist if y (0) = 0.
29. (a) Without solving, explain why the initial-value problem dy dx vy, y(xo) = yo has no solution for yo < 0. (b) Solve the initial-value problem in part (a) for yo > 0 and find the largest interval / on which the solution is defined
3. Suppose x,y,z satisfy the competing species equations <(6 - 2x – 3y - 2) y(7 - 2x - 3y - 22) z(5 - 2x - y -22) (a) (6 points) Find the critical point (0,Ye, ze) where ye, we >0, and sketch the nullclines and direction arrows in the yz-plane. (b) (6 points) Determine if (0, yc, ze) is stable. (c) (8 points) Determine if the critical point (2,0,0) is stable, where I > 0.
Solve the initial value problem ry' + xy = 1, > 0 y(1) = 2.
Consider the autonomous differential equation dy dt = = y(k - y), t> 0, k > 0 (i) list the critical points (ii) sketch the phase line and classify the critical points according to their stability (iii) Determine where y is concave up and concave down (iv) sketch several solution curves in the ty-plane.
y f(y; yo, θ) = y-0-1 where y- yo, θ > 1, and we 4. Let r be a continuous RV modeled b assume yo is a given, fixed value. Find both the MME and MLE for θ assuming a random sample of size n. This problem shows that the MME and MLE can be different. Joy
Let f(x, y) 2e-(x+y), x > 0, y > 0. Show that X, Y are independent. What are the marginal PDFS of each?
1 point) Given that y -x is a solution of dy dx2 in x > 0, find another solution yc of ус the same equation such that (xy.) is a fundamental set of solutions
5. Given y,-t is a solution of t2y" + 2t1-2y = 0, for t > 0, find a second solution y2.
6. Consider the intial value problem: 494"' + 42y + 9y = 0, y(0) = a > 0, y'(0) = -1. a. Find the solution in terms of a. Give your answer as y ... . Use x as the independent variable. Answer: b. Find the critical value of a that separate solutions that become negative from those that are always positive. critical value of a =