Math 2650 - Linear Differential Equations 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. Examples and Things to Come An ordinary differential equation is an equation which involves an unknown function and its derivatives with respect to one independent variable. (A partial differential equation involves an unknown function and its partial derivatives with respect to several independent variables.) The pendulum NiMlIm1H NiMqJiUiZEciIiMqJCUjZHRHRiUhIiI= NiMlJHBoaUc= + NiMqJiUiY0ciIiIlImxHISIi NiMqJiUiZEciIiIlI2R0RyEiIg== NiMlJHBoaUc= + NiMqKCUibUciIiIlImdHRiUlImxHISIi NiMtJSRzaW5HNiMlJHBoaUc= = F(t) NiMlIm1H NiMqJiUiZEciIiMqJCUjZHRHRiUhIiI= NiMlJHBoaUc= + NiMqJiUiY0ciIiIlImxHISIi NiMqJiUiZEciIiIlI2R0RyEiIg== NiMlJHBoaUc= + NiMqKCUibUciIiIlImdHRiUlImxHISIi NiMlJHBoaUc= = F(t) Population growth NiMqJiUiZEciIiIlI2R0RyEiIg== p = birth rate - death rate = b p -d p Radioactive decay NiMqJiUiZEciIiIlI2R0RyEiIg== q = -k q In general an NiMlIm5H th order o.d.e. is an equation of the form F( NiMlInRH , NiMlInhH , NiMqJiUjZHhHIiIiJSNkdEchIiI= , NiMqKCUiZEciIiMlInhHIiIiKiQlI2R0R0YlISIi ,..., NiMqKCklImRHJSJuRyIiIiUieEdGJyklI2R0R0YmISIi ) = 0 and a general first order o.d.e. is an equation of the form NiMqJiUjZHhHIiIiJSNkdEchIiI= = NiMtJSJmRzYkJSJ0RyUieEc= A solution of a differential equation is a differentiable function defined on some interval NiMlIklH which satisfies the equation (the o.d.e.) at every point in NiMlIklH . The solution may given explicitly or implicitly. That is this function may be defined explicitly or implicitly. An explicit solution NiMlJHBoaUc= is a solution of the form NiMtJSRwaGlHNiMlInRH = expression involving only NiMlInRH (the independent variable) and an implicit solution NiMlJFBoaUc= is of the form NiMvLSUkUGhpRzYkJSJ0RyUieEciIiE= (an expression involving both NiMlInhH and NiMlInRH , the dependent and independent variables) it implicitly defines the function NiMtJSJ4RzYjJSJ0Rw== (the solution of the differential equation). restart:with(plottools):with(plots): Warning, the name changecoords has been redefined
Consider the radioactive decay problem NiMqJiUiZEciIiIlI2R0RyEiIg== q = -k q eq1:=diff(q(t),t)=-k*q(t); NiM+JSRlcTFHLy0lJWRpZmZHNiQtJSJxRzYjJSJ0R0YsLCQqJiUia0ciIiJGKUYwISIi dsolve(eq1,q(t)); NiMvLSUicUc2IyUidEcqJiUkX0MxRyIiIi0lJGV4cEc2IywkKiYlImtHRipGJ0YqISIiRio= ic1:=q(0)=q[0]; NiM+JSRpYzFHLy0lInFHNiMiIiEmRidGKA== dsolve({eq1,ic1},q(t)); NiMvLSUicUc2IyUidEcqJiZGJTYjIiIhIiIiLSUkZXhwRzYjLCQqJiUia0dGLEYnRiwhIiJGLA== soln1:=rhs(%); NiM+JSZzb2xuMUcqJiYlInFHNiMiIiEiIiItJSRleHBHNiMsJComJSJrR0YqJSJ0R0YqISIiRio= q[0]:=2;k:=0.1; NiM+JiUicUc2IyIiISIiIw== NiM+JSJrRyQiIiIhIiI= soln1; NiMsJC0lJGV4cEc2IywkJSJ0RyQhIiJGKiIiIw== plot(soln1,t=0..100,thickness=2); 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 Consider the population growth NiMqJiUiZEciIiIlI2R0RyEiIg== p = b p - d p eq2:=diff(p(t),t)=b*p(t)-d*p(t); NiM+JSRlcTJHLy0lJWRpZmZHNiQtJSJwRzYjJSJ0R0YsLCYqJiUiYkciIiJGKUYwRjAqJiUiZEdGMEYpRjAhIiI= dsolve(eq2,p(t)); NiMvLSUicEc2IyUidEcqJiUkX0MxRyIiIi0lJGV4cEc2IyomLCYlImJHRiolImRHISIiRipGJ0YqRio= ic2:=p(0)=p[0]; NiM+JSRpYzJHLy0lInBHNiMiIiEmRidGKA== dsolve({eq2,ic2},p(t)); NiMvLSUicEc2IyUidEcqJiZGJTYjIiIhIiIiLSUkZXhwRzYjKiYsJiUiYkdGLCUiZEchIiJGLEYnRixGLA== soln2:=rhs(%); NiM+JSZzb2xuMkcqJiYlInBHNiMiIiEiIiItJSRleHBHNiMqJiwmJSJiR0YqJSJkRyEiIkYqJSJ0R0YqRio= p[0]:=2;b:=3;d:=2; NiM+JiUicEc2IyIiISIiIw== NiM+JSJiRyIiJA== NiM+JSJkRyIiIw== soln2; NiMsJC0lJGV4cEc2IyUidEciIiM= plot(soln2,t=0..1,thickness=2); 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 Consider the pendulum NiMlIm1H NiMqJiUiZEciIiMqJCUjZHRHRiUhIiI= NiMlJHBoaUc= + NiMqJiUiY0ciIiIlImxHISIi NiMqJiUiZEciIiIlI2R0RyEiIg== NiMlJHBoaUc= + NiMqKCUibUciIiIlImdHRiUlImxHISIi NiMlJHBoaUc= = 0 m:='m';c:='c';l:='l';g:=10; NiM+JSJtR0Yk NiM+JSJjR0Yk NiM+JSJsR0Yk NiM+JSJnRyIjNQ== eq3:=m*diff(phi(t),t,t)+(c/l)*diff(phi(t),t)+(m*g/l)*phi(t)=0; NiM+JSRlcTNHLywoKiYlIm1HIiIiLSUlZGlmZkc2JC0lJHBoaUc2IyUidEctJSIkRzYkRjAiIiNGKUYpKiYqJiUiY0dGKS1GKzYkRi1GMEYpRiklImxHISIiRikqJiooIiM1RilGKEYpRi1GKUYpRjpGO0YpIiIh dsolve(eq3,phi(t)); NiMvLSUkcGhpRzYjJSJ0RywmKiYlJF9DMUciIiItJSRleHBHNiMsJComKiYsJiUiY0dGKyokLSUlc3FydEc2IywmKiQpRjMiIiNGK0YrKigiI1NGKyklIm1HRjtGKyUibEdGKyEiIkYrRitGK0YnRitGKyomRkBGK0Y/RitGQSNGQUY7RitGKyomJSRfQzJHRistRi02IywkKiYqJiwmRjNGK0Y0RkFGK0YnRitGK0ZCRkFGQ0YrRis= m:=1/10;c:=0;l:=4;g:=10; NiM+JSJtRyMiIiIiIzU= NiM+JSJjRyIiIQ== NiM+JSJsRyIiJQ== NiM+JSJnRyIjNQ== eq3:=m*diff(phi(t),t,t)+(c/l)*diff(phi(t),t)+(m*g/l)*phi(t)=0; NiM+JSRlcTNHLywmLSUlZGlmZkc2JC0lJHBoaUc2IyUidEctJSIkRzYkRi0iIiMjIiIiIiM1KiYjRjMiIiVGM0YqRjNGMyIiIQ== ic3:=phi(0)=1,D(phi)(0)=0; NiM+JSRpYzNHNiQvLSUkcGhpRzYjIiIhIiIiLy0tJSJERzYjRihGKUYq dsolve({eq3,ic3},phi(t)); NiMvLSUkcGhpRzYjJSJ0Ry0lJGNvc0c2IywkKiYtJSVzcXJ0RzYjIiM1IiIiRidGMSNGMSIiIw== soln3:=rhs(%); NiM+JSZzb2xuM0ctJSRjb3NHNiMsJComLSUlc3FydEc2IyIjNSIiIiUidEdGLiNGLiIiIw== plot(soln3,t=0..15,thickness=2); 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p:=unapply(soln3,t); NiM+JSJwR2YqNiMlInRHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKC0lJGNvc0c2IywkKiYtJSVzcXJ0RzYjIiM1IiIiOSRGNSNGNSIiI0YoRihGKA== L:=(x,y)->line([0,0],[x,y],thickness=3,color=red): N:=50; NiM+JSJORyIjXQ== x:=l*sin(p((15/N)*n));y:=-l*cos(p((15/N)*n)); NiM+JSJ4RywkLSUkc2luRzYjLSUkY29zRzYjLCQqJi0lJXNxcnRHNiMiIzUiIiIlIm5HRjIjIiIkIiM/IiIl NiM+JSJ5RywkLSUkY29zRzYjLUYnNiMsJComLSUlc3FydEc2IyIjNSIiIiUibkdGMSMiIiQiIz8hIiU= seq1:=seq(L(x,y),n=0..N): display(seq1,insequence=true); 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