Math 2080 - Exam 1 - Fall 2015Printed Name: Sec:Directions: Read each question and all directions carefully before you begin.In order to receive full credit you must• Show legible and logical (relevant) justification, which supports your final answer.• Use correct and complete notation.• Simplify all answers.• Present the answer in a mathematical equation in a proper and complete English sentence.• Include proper units if appropriate.• Use techniques developed in this unit. No credit will be given for the use of any formula not derivedin class and approved by the instructor.• Show the appropriate work if you use Integration by Parts. No Credit will be given for TabularIntegration.Hint: It would be wise to check as many solutions as possible when you have finished.The use of any electronic devices (calculator, computer, cell phone, pda, etc), books, notes, or any otheraide not supplied with this exam or by the instructor is strictly prohibited.On my honor, I have neither given nor received illicit information during this exam.Signature:Problem Point Value Score1 102 103 104 105 106 10Total 601This page intentionally left blank.21. Consider the first order autonomous ode y0= y2+ y − 6.(a) Find and state the equilibrium solutions of the ode.(b) Find and state the intervals of y on which solution curves are increasing and the intervals of yon which solution curves are decreasing.(c) On one graph of the xy-plane sketch the equilibrium solutions and a possible solution curve foreach of the following initial conditions: y(0) = −5; y(0) = −1; y(0) = 1; y(0) = 3.Be sure all x-intercepts, y-intercepts and each solution curve is clearly labeled.(d) Classify each equilibrium solution as stable, unstable, or semi-stable.32. (a) Find and state the explicit solution of the ode 4y0+ 2x = x2y0.(b) Find and state the interval of existence (or domain of the solution) of the initial value problem4y0+ 2x = x2y0, y(0) = 1.Hint: You do not need to solve the ivp to answer this question.43. Find and state the explicit solution of the ode ty0− y = t3sin(t), for t > 0.54. Find and state an implicit solution of the ode (3y +1x2+ x)dx + (sec2(y) + 3x)dy = 0.65. (a) Find and state the explicit solution of the odedydx=y(x − y)x2.(b) Find and state the explicit solution of the initial value problemdydx=y(x − y)x2, y(1) =12.76. Find and state the explicit solution of the Bernoulli equation y0+yx=√y, for x >
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