Problem 2. (a) Solve the initial value problem I y' + 2y = g(t), 1 y(0)...
2y 1. (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 x, 60, b) no solution exists if y(0) = % 70, and c) an infinite number of solutions exist if y(0) = 0.
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.
x (9 points) Given the initial value problem y' 2y 29, 2014 ,y (xo) = yo. Use the existence and uniqueness theorem to show that a) a unique solution exists on any interval where Xo 70, b) no solution exists if y (0) = yo #0, and c) an infinite number of solutions exist if y (0) = 0.
(1 point) Let g(t) = e2t. a. Solve the initial value problem y – 2y = g(t), y(0) = 0, using the technique of integrating factors. (Do not use Laplace transforms.) y(t) = b. Use Laplace transforms to determine the transfer function (t) given the initial value problem $' – 20 = 8(t), $(0) = 0. $(t) = c. Evaluate the convolution integral (0 * g)(t) = Só "(t – w) g(w) dw, and compare the resulting function with the...
5. (11 points) Solve the following initial value problem, y" + 3y + 2y = g(t); y(0) = 0, 7(0) = 1/2 where g(t) = 38(t - 1) + uz(t) Type here to search
1. For the initial value problem y' = 3y2/3, y(2) = 0, there is a trivial solution, y(x) = 0. Find a nontrivial solution to this IVP. Does this contradict the existence theory for solutions of first onder IVPs y = f(x, y), y(x) = yo? Briefly explain. (VALUE: 4 l ations:
Consider the initial value problem x^2 dy/dx = y - xy, y(-1) = 1 Use the Existence and Uniqueness theorem to determine if solutions will exist and be unique. Then solve the initial value problem to obtain an analytic solution.
For each initial value problem, does Picards's theorem apply? If so, determine if it guarantees that a solutio exists and is unique. Theorem (Picard). Consider the initial value problem dy = f(t,y), dt (IVP) y(to) = Yo- (a) Existence: If f(t,y) is continuous in an open rectangle R = {(t,y) |a<t < b, c < y < d} and (to, Yo) belongs in R, then there exist h > 0 and a solution y = y(t) of (IVP) defined in...
5.Solve the initial value problem y" +5y' +6y-g(t), y(0) 0,(0) 2, where (t)-t 1<t<5,. 1, 5 < t. Then sketch the graph of the solution. (Use technologies. Be sure the graph is neat.) Sec. 7.6.39]
Find the solution of the initial value problem y" – 2y' + 5y = g(t), y(0) = 0, y'(0) = 0, where g(t) is a continuous, otherwise arbitrary, function. Oy(t) = g(t) 1 y(t) = (sets sin(2t))g(t) Oy(t) = (cos(2t)) * g(t) Oy(t) = (cos(2t))g(t) y(t) = (1 e*) + f(t) x(t) =() e sin(26)g(t) g(t) = ( e sin(2t) + (t) y(t) = Ce+ sin(2t)) *g(t) 1