Math 2650 - Qualatative AnalysisDirection FieldsA. J. MeirCopyright (C) A. J. Meir. All rights reserved.This worksheet is for educational use only. No part of this publication may be reproduced or transmitted for profit in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system without prior written permission from the author. Not for profit distribution of the software is allowed without prior written permission, providing that the worksheet is not modified in any way and full credit to the author is acknowledged.Qualitative analysisOne of the emphases of the class is qualitative analysis.This is important since most differential equations cannot be solved explicitly, although in many cases we can still say someting about the solution to the equation (existence, uniqueness, and qualatative behavior).Even thogh we may not be able to find an explicit representation for the solution we may be able to show the solution exists and that it is unique, furthermore we may be able to answer the questions:Does the solution tend to 0 or NiMlKWluZmluaXR5Rw== as NiMlInRH tends to NiMlKWluZmluaXR5Rw==?Is the solution periodic, and if so with what period?Does the solution approach 0 , or some other value only as NiMlInRH tends to NiMlKWluZmluaXR5Rw==, or for some finite time NiMlInRH?Does the solution "converege" to some other function as NiMlInRH tends to NiMlKWluZmluaXR5Rw==?Project 1Study the problems: i. NiMvKiYlI2R4RyIiIiUjZHRHISIiKiQlInhHIiIk ii. NiMvKiYlI2R4RyIiIiUjZHRHISIiLCgqJCUidEciIiNGJiokJSJ4R0YsRihGJkYma. Plot their direction field.
b. When is x(t) increasing, decreasing, constant, positive, and negative?
c. Is the solution valid for all t?
d. What are NiMtJSZsaW1pdEc2JC0lInhHNiMlInRHL0YpJSlpbmZpbml0eUc=, NiMtJSZsaW1pdEc2JC0lInhHNiMlInRHL0YpLCQlKWluZmluaXR5RyEiIg==?