exam1W09 - Name: Section/Time of lecture: Professor/GSI:...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Name: Section/Time of lecture: Professor/GSI: MATH 216 WINTER 2009 MIDTERM I To get full score you need to carefully explain what you did. No calculators allowed. 1 Problem Points Score 1 10 2 28 3 22 4 26 2 TOTAL 88 2 Problem 1. a-4pt Verify that the function y (x) = x2 sin x satisfies y ￿￿ − 3 y￿ + y = 3x cos x. x dy = x + y 2 . Include enough detail to be able dx to give a rough sketch of the solution curves with y (1) = 0 and y (0) = 0.5. b-6pt Sketch the slope field and solution curves for y 1.5 1 0.5 −2 −1.5 −1 −0.5 0.5 −0.5 −1 −1.5 4 1 1.5 2 x Problem 2. a-8pt Find the (explicit or implicit) solution for x 5 dy 2 − = 0, y (1) = π . dx cos y b-10pt Solve xy ￿ + 2y − 1 = 0, y (1) = 1. x2 6 c-10pt Let x(t) represent the number of grams of salt dissolved in a water solution. Assume that x(t) satisfies x￿ = x − x2 and x = 0.1 when t = 0 minute. How many minutes will it take for an additional 0.1 grams of salt to dissolve? 7 Problem 3. dx = x − 4x3 . Find the equilibrium solutions a-6pt Draw the phase diagram and solution curves for dt and decide whether they are stable. 8 b-10pt Let v (t) denote the speed of a motorboat at time t. The motorboat starts from rest with a constant acceleration of 10f t/sec2 and the water resistance causes a deceleration of v 2 f t/sec2 . Find the limiting velocity of the boat as t → ∞ (that is, the maximum possible speed). 9 c-6pt Use improved Euler method with stepsize 1 to estimate y (2) for 10 dy = x2 − y, y (0) = 1. dx Problem 4. a-8pt Solve 2y ￿￿ + y ￿ + y = 0, y (0) = 0, y ￿ (0) = 1 11 b-8pt Write the complex number 1 − i in polar form. c-10pt Find the general solution of the equation y (6) + 2y (4) + y ￿￿ = 0. 12 ...
View Full Document

Ask a homework question - tutors are online