math20D Stephen Young fall 09 midterm1

math20D Stephen Young fall 09 midterm1 - Math 20D Exam #1...

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Unformatted text preview: Math 20D Exam #1 19 October 2009 1. (20 points) Consider the differential equation y = (1- y )( y + 2)( y- 3) . Plot a representative set of solutions curves. Be sure to draw any equilibrium solutions and label their stability. y 3 2 1 8-1 4-2-4-8 y vs. y Solution: Since y = 0 when y =- 2 , 1 , 3 these are the equilibrium solutions. When y <- 2, y > 0, when- 2 < y < 1, y < 0, when 1 < y < 3, y > 0 and when y > 3 y < 0. Thus y =- 2 is stable, y = 1 is unstable, and y = 3 is stable. Let r 1 be the local minimum between- 2 and 1 and let r 2 be the local minimum between 1 and 3. The solution is concave up on (- 2 ,r 1 ) (1 ,r 2 ) (3 , ) and concave down on (- ,- 2) ( r 1 , 1) ( r 2 , 3).-2 t 1 0.8 0.6 0.4 y(t) 0.2 4 2 Page 1 of 3 Name: Points earned: Math 20D Exam #1 19 October 2009 2. (20 points) A tank starts off with 200 gallons of water and 25 pounds of salt in it. A fresh water solutions is pumped into the tank at a rate of sin( t ) + 5 gallons per minute, the tank is being emptied at a rate of 5- sin( t ) gallons per minute. Set up, BUT DO NOT SOLVE , the differential equation for the pounds of salt in the tank at time...
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This note was uploaded on 01/26/2010 for the course MATH 20D 20D taught by Professor Eggers,john during the Fall '09 term at UCSD.

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math20D Stephen Young fall 09 midterm1 - Math 20D Exam #1...

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