Exam1Sample - dx dt = ( x 2 + 1) 1-e-t , x (1) = 1 (Note: I...

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Name: TA: Section: Exam 1 practice test, Math 254 This is a practice test for exam 1. Please refer to the review handout for more practice problems which cover the range of all of the types of problems you are expected to be able to solve on the exam. In addition to presenting a few more practice problems, this practice test can help you to gauge that you can work through the problems in the amount of time you will have in class (about 70 minutes). In class you will have much more space to write out your solutions, I have condensed to one page to save paper if you choose to print this. On this test you must show your work . A correct solution without proper explanation will receive no credit. It is not necessary to show the steps of your integration or differentiation. 1. Solve the initial value problem dr - tan θ ( r + 1) = θ, r ± π 4 ² = π/ 8 + 2 2. Find the general solution (implicit is ok) to ( y (ln( xy ) + 1)) + ( x (ln( xy ) + 1)) y 0 = 0 (Note: this is the corrected version of the problem, I apologize for the initial error). 3. Solve the initial value problem
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Unformatted text preview: dx dt = ( x 2 + 1) 1-e-t , x (1) = 1 (Note: I xed the initial condition to make it nicer than the original value of / 4 that I had posted.). 4. Use the substitution v = y/x to solve the differential equation y = x + y x-y First calculate y in terms of x , v and v . Substitute into the differential equation and replace all expressions involving y with expressions in v and x . Solve the resulting differential equation for v (implicit form is ok). Rewrite your solution to give an implicit solution for y . 5. A tank that can hold 1000L of water starts with 500L of salt water with concentration 0.1kg/L. We pump brine with a concentration of 1kg/L (of salt) into the tank at a rate of 50L/hour. Assume that the liquid is always well mixed. (a) What will the concentration of salt be in the tank when the tank rst overows? (b) What will the concentration of salt in the tank be 15 hours after it overows?...
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