practice1(3)

# practice1(3) - when the tank is completely ±lled Recall...

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Practice Exam 1: MAP 2302 * 1. Let x be the independent, and y the dependent variable. Classify the following diFerential equations as separable, linear, exact, homogeneous, or none. Some equations may ±t in several classes, and you should list all possibilities. Do not try to solve these equations. (a) x 4 dx + ydy = 0 (b) xdx + y 4 dy = 0 (c) y 4 dx + xdy = 0 (d) ydx + x 4 dy = 0 (e) ( xy - y 2 ) dx + x 2 dy = 0 2. Solve the following equation: ( y ln( x ) + e x y 2 ) dx + ( x ln( x ) - x + 2 e x y + sin( y )) dy = 0 3. Does the following relation determine an implicit solution of the ODE? x 2 + ln( x y ) = 1 , dy dx = y + 2 x 2 y x . 4. Assume that a salty solution runs through a tank with in²ow rate 2 l/s, and out²ow rate 1 l/s. The volume of the tank is 1000 l, and initially there is no salt, and 100 l of pure water in the tank. The input concentration of salt is 1 kg/l. Determine the amount (in kg) of salt in the tank at the time
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Unformatted text preview: when the tank is completely ±lled. Recall that the ODE describing the amount of salt in the tank is: dx dt = F 1 c in ( t )-F 2 x V ( t ) , where F 1 (l/s) and F 2 (l/s) are in²ow and out²ow rates, V ( t ) (in l) is the volume of the solution at time t , and c in ( t ) (in kg/l) is the concentration of salt in the solution at the input of the tank. 5. Consider the following population model: dp dt = p (1-p ) , p (0) = p > . (a) Solve this initial value problem. (b) Sketch the direction ±eld and discuss what happens to solutions when t → ∞ . Does your answer depend on the value of p ? * Instructor: Patrick De Leenheer....
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