midterm1_07 - y p t y = c 1 g t y t = c 2 3 3[20 Suppose...

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Ordinary Differential Equations Math 22B-002, Spring 2007 Midterm 1 NAME ............................................................ SIGNATURE .................................................. I.D. NUMBER ............................................... No books, notes, or calculators. Unless stated otherwise, show all your work. Question Points Score 1 10 2 10 3 20 4 20 5 20 6 20 Total 100 1
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1. [10%] Say if the following ODEs are linear or nonlinear. By use of the theorems given in class, what can you say about the existence and uniqueness of a solution y ( t ) of the ODEs with the initial condition y (1) = - 2? (a) ( t + 1) y 0 + (cos t ) y = e t . (b) ( y + 1) y 0 + (cos t ) y = e t . 2
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2. [10%] Suppose that for certain continuous function p ( t ), g ( t ) and initial time t 0 , the functions y 1 ( t ), y 2 ( t ) are solutions of the initial value problems y 0 1 + p ( t ) y 1 = g ( t ) , y 1 ( t 0 ) = 0 , y 0 2 + p ( t ) y 2 = 0 , y 2 ( t 0 ) = 1 . If c 1 , c 2 are constants, show that the function y ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ) is the solution of the initial value problem
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Unformatted text preview: y + p ( t ) y = c 1 g ( t ) , y ( t ) = c 2 . 3 3. [20%] Suppose that a , b are constants. Find the general solution of y + ay = b. If a > 0, how does the solution behave as t → + ∞ ? 4 4. [20%] (a) Find the solution of the initial value problem ty + 3 y = 1 t , t > , y (1) = y . (b) For what initial values y is the solution y ( t ) equal to zero for some t > 0? 5 5. [20%] (a) Solve the initial value problem ty + y 2 = 0 , y (1) = 1 . (b) What is the largest t-interval in which the solution exists? 6 6. [20%] Consider the ODE y = y 2-y 4 . (a) Find all equilibrium solutions. (b) Sketch the phase line. (c) Determine the stability of the equilibria you found in (a). 7...
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