2132Final_Practice

2132Final_Practice - Polytechnic Institute of NYU MA 2132...

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Unformatted text preview: Polytechnic Institute of NYU MA 2132 Final Practice Fall 2008B Studying from past or sample exams is NOT recommended. If you do, it should be only AFTER you know how to do all of the homework and worksheet problems. These are additional practice problems designed to cover the material, but not necessarily specific to the exam. The final is cumulative, covering all material in the course. In general, 75-80% of the exam is new material. (1) Consider the initial value problem P = te t 2- P , y (0) = 0 . (a) Use the Euler Method with stepsize h = 0 . 25 to find the approximate value for the solution of the above problem at x = 0 . 5. (b) Find the exact solution and determine its value at x = 0 . 5. (2) Consider the initial value problem x 2 y = 2 xy- x 3 , y (1) = 3 . (a) Use the Euler Method with stepsize h = 0 . 1 to find the approximate value for the solution of the above problem at x = 1 . 3. (b) Find the exact solution and determine its value at x = 1 . 3. (3) Consider the initial value problem xy = y (ln( y/x ) + 1) , y (1) = e. (a) Use the Euler Method with stepsize h = 0 . 1 to find the approximate value for the solution of the above problem at x = 1 . 2. (b) Find the exact solution and determine its value at x = 1 . 2. (4) Consider the d.e. ( y 3 e- x + 2 xe y + cos( x )) dx + ( x 2 e y- 3 y 2 e- x- sin( y )) dy = 0 . (a) Show that the equation is exact. (b) Find the (implicit) solution to the d.e. (c) Solve for the particular solution satisfying y (0) = 1. (5) Find the explicit solution to the d.e. x 2 y = x 3 y 2- 2 xy, x > 0. (6) Find the general solution to the d.e. y 000- 6 y 00 + 10 y = 12 e 3 x + 50 x 2 . 1 2 (7) Find the general solution to the d.e. d 3 y dx 3- 4 dy dx = 3 e- 2 x + 2 x- cos( x ). (8) Consider the 5th order differential equation 3 d 5 y dt 5- 2 d 3 y dt 3 + 3 d 2 y dt 2- 4 y = 2 e- t (a) Express the equation as a 5-dimensional system of linear differential equations. (b) Rewrite the system in the matrix form ~ x = A~ x + ~ b . (9) Write down a fundamental set of solutions to each of the following. (a) d 3 y dx 3 = 0 (b) d 3 y dx 3- dy dx = 0 (c) d 3 y dx 3- d 2 y dx 2 = 0 (d) d 3 y dx 3- y = 0 (10) Solve the given initial value problem....
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2132Final_Practice - Polytechnic Institute of NYU MA 2132...

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