exam1 - ), W ( y 1 ,y 2 )( t ) = y 1 ( t ) y 2 ( t )-y 1 (...

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Exam I — Math 220 — September 29, 2005 Name: Directions: Show all your work on these papers, and circle your answers if that’s appropriate. 1. Solve the IVP or find the general solution if no initial conditions are given: y 0 = (1 - 2 x ) y 2 ; y (0) = - 1 / 6 ty 0 + 2 y = 5 t 3 y 00 + 6 y 0 + 9 y = 0; y (0) = 1; y 0 (0) = 1 2. A tank initially contains 200 gallons of water with 100 lbs of salt in solution. Water containing 1 pound per gallon of salt flows into the tank at a rate of 3 gal/minute. The mixture flows out of the tank at 2 gal/minute. Let Q ( t ) be the number of pounds of salt in the tank at time t . Write down, but do NOT solve, the initial value problem (DE and IC) for Q ( t ). 3. Suppose y 1 ( t ) and y 2 ( t ) are two solutions to y 00 + py 0 + qy = 0 on the interval [ a,b ]. Suppose that at some t 0 in ( a,b
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Unformatted text preview: ), W ( y 1 ,y 2 )( t ) = y 1 ( t ) y 2 ( t )-y 1 ( t ) y 2 ( t ) 6 = 0. Show that the initial value problem y 00 + py + qy = 0; y ( t ) = y , y ( t ) = y has a unique solution in the form y ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ). 1 4. Find the general solution to y 00-y + y = cos(2 t ) . 5. Using Eulers method, with t = 1 / 2, nd an approximate value for y (1), where y + 4 y = t ; y (0) = 1. 6. For the autonomous DE dx dt = 1-x 2 , identify the critical points and classify them as stable or unstable. Plot (qualitatively) some representative solutions (dont forget to include the equilibrium solutions) in the t-x plane below. 2...
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exam1 - ), W ( y 1 ,y 2 )( t ) = y 1 ( t ) y 2 ( t )-y 1 (...

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