practice final

practice final - all real x . 6. Consider the system of...

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Math 364 Ordinary Differential Equations Practice Final Exam Show your work. Be complete and precise. 1. Find the solution to the initial value problem ¢ ¢ y + y = sin x , y (0) = 0, ¢ y (0) = 1. 2. Find the general solution to the differential equation ¢ ¢ y + 2 ¢ y + y = e x . 3. Find two linearly independent solutions to the equation ¢ ¢ y - 2 x ¢ y + 6 y = 0. 4. Prove the following theorem. If f 1 ,..., n are n solutions of y ( n ) + a 1 ( x ) y ( n - 1) + ... + a n ( x ) y = 0 on an interval I, they are linearly independent there if, and only if, W ( 1 ,..., n )( x ) 0, for all x in I, where W ( 1 ,..., n )( x ) is the Wronskian of 1 ,..., n . 5. Consider the system of equations ¢ y 1 = 3 y 1 + xy 3 ¢ y 2 = y 2 + x 3 y 3 ¢ y 3 = 2 xy 1 - y 2 + e x y 3 . Show that every initial value problem for this system has a unique solution which exists for
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Unformatted text preview: all real x . 6. Consider the system of equations y 1 = -3 y 1 + 4 y 2 y 2 = -2 y 1 + 3 y 2 . a) Find the general solution of this system. b) Find the solution to the initial value problem given by this system and y 1 (0) = 1, y 2 ( x ) = 0. 7. A large tank contains 100 gallons of brine in which 200 pounds of salt are dissolved. Beginning at time t = 0 pure water runs in at the rate of 3 gallons/minute, and the mixture, which is kept uniform by stirring, flows out at the rate of 2 gallons/minute. How long will it take to reduce the amount of salt in the tank to 100 pounds?...
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