Computer Solved Differential Equations

By Joe J.

This book is a how-to-do booklet. The differential equations have the answers in tabular form. Knowing how to use a table of derivatives is necessary. The computer language was in basic format. You can use any computer language to do these problems. The author has 35 years of teaching in a technical college setting. The courses taught were mechanical engineering, physics, and math courses. The differential equations course was taught using classroom lectures and a lab session using experiments. The student had to present a report containing classical solutions, experimental results, and computer solutions to each physical experiment.

• Language: English
• Publisher: Xlibris US
• Release date: May 21, 2012
• ISBN: 9781477112274

Joe J.

Here is the timeline of my experience. B.S. in General Engineering in 1955 from Trinity College in Hartford, Ct. The U.S. army from 1955 to 1957 as a Nike Guided Missile Analyst. An engineer from 1957 to 1958. Master's Degree from University of Connecticut in 1960. An engineer from 1960 to 1961. Technical College teaching from 1961 to 1963. An engineer from 1963 to 1964. Technical College teaching from 1964 to 1995. Tutoring in math from 1995 to the present time.

Book preview

Copyright © 2012 by Joe J. Ettl.

All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the copyright owner.

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116419

Contents

PREFACE

D² y = 6x + 2

D³ y = 24x + 6

D² y = sin x

D² y = ex

D³ y = 24x − 24

D³ y = 24x −12

D³ y = 48x − 30

D² y = 6 − 6x

D² y = −24x

D² y = 384x − 72

D⁵ y = 720x

Dy = −y

Dy = y

Dy = 1− y

Dy + y = 2cost

Dy = 1+ x + y

Dy = 1+ x − y

Dy = x² + y

Dy = x cos x + y

D² y = 0

D² y = 1

D² y = −1

D² y = 32

D² y = 32 − Dy

D² y = −36y

D² y + 6Dy + 25y = 0

D² y +10Dy + 25y = 0

D² y +14.5Dy + 25y = 0

D² y + 2Dy + 2y = cost − 2sin t

D² y + y = 8cos3t

D² y + y = 2cos t

D⁴ y +10D² y + 9y = 0

10DQ + 50Q = 20

10i + 0.1Di = 50

116419-ETTLa.jpg116419-ETTL-layout.pdf116419-ETTL-layout.pdf116419-ETTL-layout.pdf116419-ETTL-layout.pdf116419-ETTL-layout.pdf

Dy = y − y²

Dy = y² − y

Comments and References

This booklet is dedicated to my wife.

PREFACE

The purpose of this booklet is to show you a computer solution is a good approximation to the analytical solution of a linear differential equation with constant coefficients.

I used a pocket scientific computer where programs were written in basic language. An end statement was not used in any program because the program stopped each time a print statement was executed. After each print statement I had to press the enter key to continue the program. I found the error between the computer and the analytical results was in the tenth significant figure most of the times and sometimes in the ninth significant figure.

You are expected to know how to use a table of derivatives. The symbol Dy means the first derivative of y with respect to the independent variable. The symbol missing image file means the nth derivative of y with respect to the independent variable. I used these symbols to make the typing easier. The linear differential equation with constant coefficients is given as:

116419-ETTL-layout.pdf

Note: A and B are constants.

You will find most of the differential equations deal with positive values of the independent variable and start with zero. The applications deal with practical problems.

The computer program was written to follow the solution of differential equations by use of series. As an example of this process, we will start with Dy = y where y = 1 when x = 0. The analytical solution is missing image file and it’s series expansion is as follows:

116419-ETTL-layout.pdf

The computer

rocess is given by

116419-ETTL-layout.pdf

,

116419-ETTL-layout.pdf which leads to . 116419-ETTL-layout.pdf

If we start with dy12 = ydx, we have after much algebra

116419-ETTL-layout.pdf

. I have used 12 terms in all the solutions even if it was not necessary.

The solution of the upcoming differential equation is missing image file .

The differential equation is missing image file where y = 0 and Dy = 0 when x = 0.

Step 1.) 116419-ETTL-layout.pdf

Step 2.) 116419-ETTL-layout.pdf

Step 3.) Computer Program