Math 2650 A. J. Meir Copyright (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. restart:with(DEtools):with(plots):

Consider the equation 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRidGPkY+LUkjbW9HRiQ2LVEiPUYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkkvJSlzdHJldGNoeUdGSS8lKnN5bW1ldHJpY0dGSS8lKGxhcmdlb3BHRkkvJS5tb3ZhYmxlbGltaXRzR0ZJLyUnYWNjZW50R0ZJLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGWC1GOzYkUSIxRidGPkZARj4= We verify that y:= x-> 1/(1-x);

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQsJiIiIkYrOSQhIiJGLUYlRiVGJQ==

is a solution. On the interval (0, 1), it is a differenitiable function, and D(y)(x)-y(x)^2;

IiIh

so it satisfies the equation, and y(0);

IiIi

so it satisfies the initial condition, hence it is a solution. Lets try and use Euler's method to construct an approximate solution Below we write a little Maple procedure for the forward Euler's method.

FEuler:=proc(f,t0,x0,h,N) local t,x,n,sol: t:=evalf(t0): x:=evalf(x0): sol:=[[t,x]]: for n from 1 to N do x:=x+h*f(t,x): t:=t+h: sol:=[op(sol),[t,x]]: od; end;

Zio2J0kiZkc2IkkjdDBHRiVJI3gwR0YlSSJoR0YlSSJOR0YlNiZJInRHRiVJInhHRiVJIm5HRiVJJHNvbEdGJUYlRiVDJj44JC1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjOSU+OCUtRjM2IzkmPjgnNyM3JEYxRjg/KDgmIiIiRkI5KEkldHJ1ZUdGNEMlPkY4LCZGOEZCKiY5J0ZCLTkkNiRGMUY4RkJGQj5GMSwmRjFGQkZJRkI+Rj03JC1JI29wR0Y0NiNGPUY/RiVGJUYl

y:='y';

SSJ5RzYi

x0:=0;xf:=1.2;y0:=1;

IiIh

JCIjNyEiIg==

IiIi

N:=6;

IiIn

h:=evalf((xf-x0)/N);

JCIrKysrKz8hIzU=

f:=(x,y)->y^2;

Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQ5JSIiI0YlRiVGJQ==

out:=FEuler(f,x0,y0,h,N);

Nyk3JCQiIiFGJSQiIiJGJTckJCIrKysrKz8hIzUkIisrKysrNyEiKjckJCIrKysrK1NGKyQiKysrKylbIkYuNyQkIisrKysrZ0YrJCIrKylHMyQ+Ri43JCQiKysrKyshKUYrJCIrcihba24jRi43JCQiKysrKys1Ri4kIit3Vjc0VEYuNyRGLCQiKy9dNSdbKEYu

EulerApprox:=plot(out,color=green,thickness=2): DF:=DEplot(diff(y(x),x)=y(x)^2,y(x),x=-1..1.5,y=0..8): EX:=plot(1/(1-x),x=0..1,color=blue,thickness=2): display(EulerApprox,DF,EX);

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Note the exact solution (blue) has a vertical asymptote at x=1. The approximate solution (green) misses this feature, and suggests that the solution is valid for values of x greater than 1. Note also the mismatch between the slope lines and the approximate solution. This is of course wrong, and is due to the finite step size. Now for 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 we will use Euler's method again. First with a stepsize h=0.1 x0:=0;xf:=1;y0:=1;

IiIh

IiIi

IiIi

N:=10;

IiM1

h:=evalf((xf-x0)/N);

JCIrKysrKzUhIzU=

f:=(x,y)->x^2+y;

Zio2JEkieEc2IkkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJDkkIiIjIiIiOSVGLkYlRiVGJQ==

out01:=FEuler(f,x0,y0,h,N);

Ny03JCQiIiFGJSQiIiJGJTckJCIrKysrKzUhIzUkIisrKysrNiEiKjckJCIrKysrKz9GKyQiKysrKzY3Ri43JCQiKysrKytJRiskIisrKzVPOEYuNyQkIisrKysrU0YrJCIrKytyeTlGLjckJCIrKysrK11GKyQiKys1ZVU7Ri43JCQiKysrKytnRiskIisrIlI9JD1GLjckJCIrKysrK3FGKyQiKzVJLV4/Ri43JCQiKysrKyshKUYrJCIrNmA3MEJGLjckJCIrKysrKyEqRiskIitVeWoqZiNGLjckJEYqRi4kIitFO2dTSEYu

Next with a stepsize h=0.05 N:=20;

IiM/

h:=evalf((xf-x0)/N);

JCIrKysrK10hIzY=

out005:=FEuler(f,x0,y0,h,N);

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

O1:=plot(out01,color=green,thickness=2): O05:=plot(out005,color=blue,thickness=2): display(O1,O05);

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

The accuracy of the approximate solution (the numerical solution) increases as the step size decreases.