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C Roettger, Fall 13 Math 267 - Practice exam 1 State clearly what your result is. Show your work: When you introduce variables, write down what they mean. For word problems, write down equations expressing the information given. No notes or books. Time is 50 minutes. Give exact results where possible, no decimal approximations. Problem 1 Have a look at the whole exam paper. Which of the numbered differential equations is/are linear? (just list their numbers). Problem 2 For which of the following initial value problems does Theorem 1.2.1 (section 1.2 of Chapter 1) guarantee existence and uniqueness of a solution, at least in a small rectangle? Justify your answers. √ y(0) = 2 (1) y 0 = √4 − x2 + |y| y(2) = 0 (2) y 0 = √4 − x2 + |y| 3 0 2 y(2) = 0 (3) y = p4 − x 0 2 y = 4 − y + |x| y(0) = 1 (4) Problem 3 Consider the differential equation 2 xy 0 + 2x2 y = (x + 5)e−x . (5) a) Find the general solution. Write down intermediate steps explaining the method you are using - the result alone is not enough. b) Solve the initial value problem consisting of (5) with initial condition y(1) = 2e−1 . c) What can you say about existence and uniqueness of solutions of an initial value problem (IVP) involving (5) and a condition y(a) = b? What is the biggest interval where you can be sure that solutions of such an IVP are defined? Problem 4 Consider the differential equation y 3 + cos(x)y 4 + [4y 3 + 4 sin(x)y 3 + 3xy 2 ] 1 dy = 0. dx (6) a) Show that this equation is exact by performing the test for exactness. b) Find an implicit solution of equation (6). Problem 5 Which of the following differential equations belongs to the direction field plotted below? p0 = p(2 − p), p0 = p2 − 3p, p0 = p2 − 4p. (7) (8) (9) For initial conditions P (0) = 0, 1, 2, 3, 3.9, 4, 4.1, make a rough sketch of solution curves. Problem 6 Consider the differential equations p a) y 0 = x + y − 1 b) xy 0 + 3y = 6xy 4 c) (x − y)y 0 = 2x + y (10) (11) (12) Give substitutions v = v(x, y) for each of them transforming them into separable or linear equations. Write down the definition of v and the differential equation for v which is separable/linear - do NOT solve them. Problem 7 A leaf floats gently down from a tree. Its vertical velocity v obeys the equation dv = −δv − g (13) dt 2 where −g = −10m/s2 is the acceleration due to gravity and the drag coefficient δ = 5 models air resistance. The tree is 20 meters high and the leaf has velocity zero initially. a) Find the solution of this initial value problem for v(t). b) Integrate once more to find the position at time t. c) Bonus: estimate how long it takes for the leaf to reach the ground. Hint – compute the height above ground at t = 10 and t = 10.5 seconds. 3