Computer Solved Differential Equations

Copyright © 2012 by Joe J. Ettl.

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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

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:

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:

The computer

rocess is given by

,

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

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

. 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