exam2_practice

# Exam2_practice - F warms up to 40 F in 3 min while sitting in a room of tem-perature 70 F Write the diﬀerential equation with all initial data

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Preparation to Exam 2. 1. Complete homework. 2. Fix all quiz mistakes. 3. (a) Verify that y 1 = x and y 2 = x ln x are two linearly independent solutions of the diﬀerential equation x 2 y 00 - xy 0 + y = 0 on (0 , ) and then write a general solution to this equation. (b) Now consider the equation x 2 y 00 - xy 0 + y = 0 on ( -∞ , ). Determine for which values x 0 , y 0 , y 1 , the Existence-Uniqueness theorem guarantees existence and unique- ness of a solution to the IVP with the initial conditions y ( x 0 ) = y 0 , y 0 ( x 0 ) = y 1 and also determine the largest interval on which existence of the solution is guaranteed. (c) Find by inspection a solution to the IVP x 2 y 00 - xy 0 + y = 0, y (0) = 0, y 0 (0) = 0. 4. Solve the following (a) y 00 - 3 y 0 + 5 y = 0 (b) y 00 - 2 y 0 + y = 0, y (0) = 5, y 0 (0) = 10 (c) y 00 + 6 y 0 + 5 y = 0, y (0) = 0, y 0 (0) = 3 5. (a) Does the IVP ( y 0 ( x ) = x 2 + cos( y ( x )) y (1) = 0 have a unique solution? Justify. (b) Find the value of the second derivative of this solution at point x = 1. (c) Show that every solution curve for y 0 ( x ) = x 2 + cos( y ( x )) increases on ( -∞ , - 1). 6. A cold beer initially at 35
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Unformatted text preview: F warms up to 40 F in 3 min while sitting in a room of tem-perature 70 F . Write the diﬀerential equation with all initial data for this problem. Find the formula for the temperature of the beer as a function of time. 7. A swimming pool whose volume is 10 , 000 gal contains water that is 0 . 01% chlorine. Starting at t = 0, city water containing 0 . 001% chlorine is pumped into the pool at a rate 5 gal/min. The pool water ﬂows out at the same rate. Set up the IVP for the unknown x ( t ), the volume of chlorine in the pool at time t . 8. Set up the IVP for the motion of the falling body under the forces of gravity and air resistance, where we assume that the air resistance is proportional to the velocity, in case the initial velocity is directed upward and the initial speed equals p . 1...
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## This note was uploaded on 12/12/2011 for the course MAP 2302 taught by Professor Katsevich during the Fall '08 term at University of Central Florida.

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