Exam 1 - Fall 2003 - Brine containing pound of salt per...

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Differential Equations Exam #1 September 18, 2003 Name ________________________________ Show all your work neatly and in numerical order on notebook paper. DO NOT WRITE ON THE BACKS OF YOUR PAGES. 1. Given 0 = + x x : (a) Show that t c t c x sin cos 2 1 + = is a two-parameter family of solutions for the equation. (b) Find a particular solution satisfying the initial conditions ( 29 1 2 / , 0 ) 2 / ( = = π x x . 2. Solve: ( 29 dy x y dx y 2 2 sec 1 = + 3. Solve: ( 29 1 1 2 = + + y y x 4. Solve: ( 29 x y dx dy x ln 1 = + + subject to the initial condition 10 ) 1 ( = y 5. Solve the exact equation: ( 29 ( 29 0 8 4 4 5 3 = - + + dy y x dx y x 6. Solve the Bernoulli equation: 2 y e y dx dy x = - 7. The population of bacteria in a culture grows at a rate proportional to the number of bacteria present at time t. After 3 hours it is observed that there are 400 bacteria present. After 10 hours there are 2000 bacteria present. What was the initial number of bacteria? You must start by writing a differential equation and solving it. 8. A large tank is partially filled with 100 gallons of fluid in which 10 pounds of salt is dissolved.
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Unformatted text preview: Brine containing pound of salt per gallon is pumped into the tank at a rate of 6 gal/min. The well-mixed solution is then pumped out at a slower rate of 4 gal/min. Find the number of pounds of salt in the tank after 30 minutes. 9. A tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its bottom. Suppose the tank is 20 feet high and has radius 8 feet and the circular hole has radius 2 inches. The differential equation governing the height h of water leaking from the tank is 2 / 3 6 5 h dt dh-= . In this model, friction and contraction of the water at the hole were taken into account with 6 . = c and 2 / 32 s ft g = . If the tank is initially full, how long will it take the tank to empty?...
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