4100_F09_E1_practice - R te& 2 t dt =& t= 2 e& 2...

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Math 4100 Practice Exam I Dr. Ciprian Gal Fall 2009 Remark: The actual exam may contain similar problems and questions that do not appear here. However, this practice should give you an idea of the level of di¢ culty that one should expect from one-hour exam. The actual exam may contain a less number of problems. 1. Solve the initial value problem ty 0 + y = t 3 ; y (1) = 3 : 2. Find the general solution of the di/erential equation y 00 + 6 y 0 + 25 y = 0 : 3. Solve the di/erential equation 2 x + y 3 + (3 xy 2 + 1) dy dx = 0 4. Solve the initial value problem y 00 + y 0 ° 6 y = 0 ; y (0) = 0 ; y 0 (0) = 1 : 5. The function y 1 ( t ) = e t is a solution of the ODE ( t + 1) y 00 ° y 0 ° ty = 0 ; t > 0 : Find the general solution. Hint: You may use the following fact: the derivative of
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Unformatted text preview: R te & 2 t dt = & ( t= 2) e & 2 t & (1 = 4) e & 2 t . 6. Solve the initial value problem y 3 + x 2 dy dx = 0 ; y (1) = 1 : 7. A stirred tank contains 300 gallons of water and 5 pounds of salt. A solution of water and salt containing 1±10 pound of salt per gallon is pumped into the tank at the rate of 15 gal±min and the mixture is allowed to drain at the same rate. Set up the initial value problem (ODE plus initial conditions) for the amount Q of salt in the tank at time t . Then, solve the initial value problem and interpret your solution. What is the behavior of salt concentration as time t ! + 1 ?...
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