# midterm1fall06.pdf

## Mathematics 216 with Epstein at University of Michigan - Ann Arbor *

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Name: Section/Time of lecture: Professor/GSI: MATH 216 MIDTERM I Each part of a problem counts equally. To get full score you need to carefully explain what you did. No calculators allowed. 1 Problem Points Score 1 30 2 30 3 30 4 60 TOTAL 150 2 Problem 1. a) Verify that the function y(x) = 1x + lnx satisfies x2yprimeprime + 2xyprime −xy = −xlnx b) Solve dydx = sinx(cosx)2 + 1x2,y(2pi) = 1. c) Sketch the slope field and solution curves for dydx = √1−x2 where defined. Include enough detail to be able to give a rough sketch of the solution curves with y(0) = 0 and y(0) = 1. 3 Problem 2. a) Solve dydx = x(1−2y)e2x,y(0) = 2 b) Solve (x2 + 1)yprime = −3xy + x,y(0) = 0. c) The time rate of change of a population P is proportional to the cube root of P. At time t=0, the population is 1000 and the rate of increase is 100 at that time. What is the population at time t=10? 4 Problem 3. a) Let dxdt = −5x + 2x3. Find the equilibrium solutions. Which ones are stable? b) Suppose that a motor boat is moving at 100 m/s when the motor quits. After 20 seconds, the boat has slowed to 10 m/s. Assume the resistance is proportional to its speed. Hence we have the differential equation for the speed v : dv dt = −kv. How far will the boat coast? c) Use Euler with stepsize .5 to estimate y(2) for dydx = x2 + y3,y(1) = 0. 5 Problem 4. a) Solve the equation yprimeprime −3yprime −10 = 0 with initial conditions y(0) = 3,yprime(0) = −1. b) Solve the equation 5z2 −2z + 10 = 0. c) Solve 9yprimeprime + 6yprime + 25y = 0,y(0) = 2,yprime(0) = 1 6

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