Differential Equations Math 285 Hour Exam 1 Fall, 2011

1.	Determine the type of each of the following equations. Possible answers are ‘separable’, ‘linear’, ‘Bernoulli’, ‘homogeneous’, ‘exact’, or ‘none’. To explain your answers, convert each equation to the form in which it is easily recognized. a. (x^2 – xy) dx + xy e^x/ydy = 0 b. e^x y’ – 2 cos x + 3 xy + 4 = 0 c. ( y – x^2 y) dx + e^x-3ydy = 0 d. (2 + 3y^3 sin x) dx + 4 x^2y^2 dy = 0 Answers: a. homogeneous b. Linear c. Separable d. Bernoulli
2.	Solve the following initial value problem. x y’ + ( 1 + 2x) y = e^x y(1) = 0 Answer: y(x) = (1 / 3x) (e^x - e ^3-2x )
3.	Find a general solution of the following homogeneous equation and express y in terms of x. x^2 y’ = x y + y^2 e^–x/y Answer: y = x / [ ln ( - ln| x | - C ]
4.	A tank initially contains 50 liters of pure water. A mixture containing 2 g/liter of salt enters the tank at a rate of 10 liters/min and the well stirred mixture leaves the tank at the same rate. a. Find the amount of salt in the tank after t minutes. b. When will the concentration of salt in the tank reach 1 g/liter? Answers: a. x = 100 ( 1 – e ^–t/5 ) b. t - 5 ln 2 Click here to continue with this exam.

Return to "Hints and Answers to Exams"

Return to Table of Course Notes