Differential Equations Math 285 Practice Hour Exam 1 Fall, 2011

1.	Without solving the equation (y-2) y’ = x + e ^y + 1 answer the following questions. a. Does the existence/uniqueness theorem guarantee the existence of a unique solution to the equation with the initial condition y(2) = 0? b. Same question for y(0) = 2. c. Find the equation of the tangent line to the graph of the solution to (a) at the point (2,0). Answers: a. Yes b. No c. y = - 2x + 4
2.	Determine the type of each of the following equations. a. xy^-2 dy – (y^-1e^x + y ln x) dx = 0 b. (2xy^3 – 2y^4) dx + (3x^2y2 – 8xy^3) dy = 0 c. xy^2y’ = y sin x + cos x d. xy^2 dx + (x + y)^2 √(x^2 - y^2) dy = 0 e. e^x-yy’ + xy^2 = y cos x The possible answers are: separable, linear, Bernoulli, homogeneous, exact, or none. To explain your answers, convert each equation to the form so it is easily recognized. Answers: a. Bernoulli b. Exact c. None d. Homogeneous e. Separable
3.	Solve the initial value problem for y’ = 2xy + 3x^2 exp(x^2), y(0) = 5. Answer: y = (x^3 + 5) exp(x^2)
4.	Solve the initial value problem y’ = x(4 – x^2)/y^3, y(1) = - 2 Express y in terms of x and find the biggest single interval in which the solution is defined. Answer: y(x) = - ( - x^4 + 8 x^2 + 9 )1/4 defined for - 3 < x < 3
5.	Find a general solution of the equation x(x + y) y’ = y(3x + y). Answer: ye ^y/x = Cx^3
6.	A bullet enter a 10 cm wide board at 250 m/sec and leaves the board at 50 m/sec. How long does it take the bullet to go through if the resistance is proportional to the cube of the speed? Answer: 0.0012 sec

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