Math 2650A. 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.

ConsiderNiMvKiYlI2R4RyIiIiUjZHRHISIiLSUiZkc2JCUidEclInhH NiMvLSUieEc2IyYlInRHNiMiIiEmRiVGKQ==A solution of the initial value problem is a continuous function NiMtJSRwaGlHNiMlInRH defined on some interval NiMlIklH containing NiMmJSJ0RzYjIiIh such that1. NiMvLSUkcGhpRzYjJiUidEc2IyIiISYlInhHRik=.2. NiMtJSRwaGlHNiMlInRH satisfies the differential equation at all but a finite number of NiMlInRH-values in NiMlIklH (that is NiMlJHBoaUc= is piecewise differentiable on NiMlIklH).

ConsiderNiMvKiYlI2R4RyIiIiUjZHRHISIiLSUiZkc2JCUidEclInhH NiMvLSUieEc2IyYlInRHNiMiIiEmRiVGKQ==Assume NiMtJSJmRzYkJSJ0RyUieEc= is a continuous function of NiMlInRH and NiMlInhH in some region R with (NiQmJSJ0RzYjIiIhJiUieEdGJQ==) interior to R,then the initial value problem has a solution (at least one solution) on some interval NiMlIklH containing NiMmJSJ0RzYjIiIh.

ConsiderNiMvKiYlI2R4RyIiIiUjZHRHISIiLSUiZkc2JCUidEclInhH NiMvLSUieEc2IyYlInRHNiMiIiEmRiVGKQ==Assume that the initial value problem has a solution NiMvJSJ4Ry0lJHBoaUc2IyUidEc=. In addition assume that NiMtJiUiZkc2IyUieEc2JCUidEdGJw== is a continuous function of NiMlInRH and NiMlInhH in some region R with (NiQmJSJ0RzYjIiIhJiUieEdGJQ==) interior to R, then the initial value problem has a unique solution (at most one solution) through (NiQmJSJ0RzYjIiIhJiUieEdGJQ==). That is, NiMvJSJ4Ry0lJHBoaUc2IyUidEc= is the unique solution through (NiQmJSJ0RzYjIiIhJiUieEdGJQ==).

Consider the equationWhen I do it I get:You should also look at the graphs of the solutions: with(DEtools):with(plots):

Warning, the name changecoords has been redefined

NiMvKiYlI2R4RyIiIiUjZHRHISIiLCQtJSVzcXJ0RzYjJSJ4R0Yo NiMvLSUieEc2IyIiISIiIg==.Lets define the differential equation (corresponding to the above equation):Lets solve the equation using the dsolve command. You can (in fact should) also try solving the equation by hand.de:=D(x)(t)=-sqrt(x(t));

NiM+JSNkZUcvLS0lIkRHNiMlInhHNiMlInRHLCQqJC0lJXNxcnRHNiMtRipGKyIiIiEiIg==

sol:=dsolve({de,x(0)=1},x(t));

NiM+JSRzb2xHLy0lInhHNiMlInRHLSUnUm9vdE9mRzYjLCgqJC0lJXNxcnRHNiMlI19aRyIiIiIiI0YpRjNGNCEiIg==

Well, how do we make sense out of this? Lets solve this equation by hand and see what we get. NiMvLSUieEc2IyUidEcsKComKiQpRiciIiMiIiJGLSIiJSEiIkYtRidGL0YtRi0= and NiMvLSUieEc2IyUidEcsKComKiQpRiciIiMiIiJGLSIiJSEiIkYtRidGLUYtRi0=.plot([t^2/4-t+1,t^2/4+t+1],t=-4..4);

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

State the existence and uniqueness theorems. Does the initial value problem satisfy these hypotheses?Are these solutions valid (verify that these are indeed solutions, are both valid solutions, or is only one of these a valid solution)? Explain.Hint: study the direction field for this equation.For what values of NiMlInRH are these solutions valid? Explain.Think about this problem, what do you conclude.