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.

Consider the differential equationNiMsJi0lIk1HNiQlInRHJSJ4RyIiIi0lIk5HRiZGKQ==NiMvKiYlI2R4RyIiIiUjZHRHISIiIiIhor (equivalently) written in differential form the equationNiMvLCYqJi0lIk1HNiQlInRHJSJ4RyIiIiUjZHRHRitGKyomLSUiTkdGKEYrJSNkeEdGK0YrIiIh.Theorem. Let the functions NiMlIk1H, NiMlIk5H, NiMmJSJNRzYjJSJ4Rw==, NiMmJSJORzYjJSJ0Rw== (where subscripts denote partial derivatives) be continuous in a simply connected region NiMlIlJH (of the NiMsJiUidEciIiIlInhHISIi plane). The above differential equation is an exact differential equation in NiMlIlJH if and only ifNiMvLSYlIk1HNiMlInhHNiQlInRHRigtJiUiTkc2I0YqRik=at each point of NiMlIlJH. That is, there exists a function NiMlJFBoaUc= satisfyingNiMvLSYlJFBoaUc2IyUidEc2JEYoJSJ4Ry0lIk1HRik= and NiMvJiUkUGhpRzYjJSJ4Ry0lIk5HNiQlInRHRic=if and only if NiMlIk1H and NiMlIk5H satisfy NiMvLSYlIk1HNiMlInhHNiQlInRHRigtJiUiTkc2I0YqRik=.If the equationNiMvLCYqJi0lIk1HNiQlInRHJSJ4RyIiIiUjZHRHRitGKyomLSUiTkdGKEYrJSNkeEdGK0YrIiIhis exact then an explicit solution is given byNiMvLSUkUGhpRzYkJSJ0RyUieEclImNHwhere NiMlJFBoaUc= satisfies NiMvLSYlJFBoaUc2IyUidEc2JEYoJSJ4Ry0lIk1HRik= and NiMvJiUkUGhpRzYjJSJ4Ry0lIk5HNiQlInRHRic=.This implicit solution may be constructed as follows:NiMvLSUkUGhpRzYkJSJ0RyUieEcsJi0lJGludEc2JC0lIk1HRiZGJyIiIi0lImhHNiNGKEYvandNiMvLSYlJFBoaUc2IyUieEc2JCUidEdGKCwmLSUlZGlmZkc2JC0lJGludEc2JC0lIk1HRilGKkYoIiIiKiYlImRHRjQlI2R4RyEiIkY0NiMtJSJoRzYjJSJ4Rw===NiMsJi0lJGludEc2JC0mJSJNRzYjJSJ4RzYkJSJ0R0YrRi0iIiIqJiUiZEdGLiUjZHhHISIiRi4=NiMtJSJoRzYjJSJ4Rw==which yieldsNiMqJiUiZEciIiIlI2R4RyEiIg==NiMvLSUiaEc2IyUieEcsJi0lIk5HNiQlInRHRiciIiItJSRpbnRHNiQtJiUiTUdGJkYrRiwhIiI=henceNiMvLSUkUGhpRzYkJSJ0RyUieEcsJi0lJGludEc2JC0lIk1HRiZGJyIiIi1GKzYkLCYtJSJOR0YmRi8tRis2JC0mRi42I0YoRiZGJyEiIkYoRi8=.

1. Consider(NiMsJiomJSJ4RyIiIi0lJGNvc0c2IyUidEdGJkYmKigiIiNGJkYqRiYtJSRleHBHNiNGJUYmRiY=)+(NiMsKC0lJHNpbkc2IyUidEciIiIqJilGJyIiI0YoLSUkZXhwRzYjJSJ4R0YoRihGKCEiIg==)NiMvKiYlI2R4RyIiIiUjZHRHISIiIiIhwe first check that this is in fact an exact differential equationdiff(x*cos(t)+2*t*exp(x),x);

NiMsJi0lJGNvc0c2IyUidEciIiIqKCIiI0YoRidGKC0lJGV4cEc2IyUieEdGKEYo

diff(sin(t)+t^2*exp(x)-1,t);

NiMsJi0lJGNvc0c2IyUidEciIiIqKCIiI0YoRidGKC0lJGV4cEc2IyUieEdGKEYo

so it is exact. Now lets calculate the solutionint(x*cos(t)+2*t*exp(x),t);

NiMsJiomJSJ4RyIiIi0lJHNpbkc2IyUidEdGJkYmKiYpRioiIiNGJi0lJGV4cEc2I0YlRiZGJg==

soNiMvLSUkUGhpRzYkJSJ0RyUieEcsKComRigiIiItJSRzaW5HNiNGJ0YrRisqJkYnIiIjLSUkZXhwRzYjRihGK0YrLSUiaEdGM0YrandNiMqJiUiZEciIiIlI2R4RyEiIg==NiMvLSUiaEc2IyUieEcsKi0lJHNpbkc2IyUidEciIiIqJkYsIiIjLSUkZXhwR0YmRi1GLUYtISIiLCZGKUYtRi5GLUYysoNiMqJiUiZEciIiIlI2R4RyEiIg==NiMvLSUiaEc2IyUieEcsJCIiIiEiIg==andh(x)=-x.Therefore an implicit solution is given byNiMvLCgqJiUieEciIiItJSRzaW5HNiMlInRHRidGJyomRisiIiMtJSRleHBHNiNGJkYnRidGJiEiIiUiY0c=.

Determine whether each of the equations is exact, if it is find the solution.1. (NiMsJiomIiIjIiIiJSJ4R0YmRiYiIiRGJg==)+(NiMsJiomIiIjIiIiJSJ5R0YmRiZGJSEiIg==)NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh2. (NiMsJiomIiIjIiIiJSJ4R0YmRiYqJiIiJUYmJSJ5R0YmRiY=)+(NiMsJiomIiIjIiIiJSJ4R0YmRiYqJkYlRiYlInlHRiYhIiI=)NiMvKiYlI2R5RyIiIiUjZHhHISIiIiIh3. NiMvLCYqJiwoKiYiIiQiIiIqJCUidEciIiNGKUYpKihGLEYpRitGKSUieEdGKSEiIkYsRilGKSUjZHRHRilGKSomLCgqJiIiJ0YpKiRGLkYsRilGKUYqRi9GKEYpRiklI2R4R0YpRikiIiE=4. NiMvLCYqJiwmKigiIiMiIiIlInRHRiklInhHRihGKSomRihGKUYrRilGKUYpJSNkdEdGKUYpKiYsJiooRihGKSokRipGKEYpRitGKUYpKiZGKEYpRipGKUYpRiklI2R4R0YpRikiIiE=