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. Laplace Transforms In order to use Laplace transforms we must load the integral transforms package. restart:with(inttrans): We compute some transforms of functions and some inverse transforms laplace(sinh(t),t,s); NiMqJCwmKiRJInNHNiIiIiMiIiIhIiJGKUYq invlaplace(1/(s^2-1),s,t); NiMtSSVzaW5oRzYkSSpwcm90ZWN0ZWRHRiZJKF9zeXNsaWJHNiI2I0kidEdGKA== laplace(t^2*sinh(t),t,s); NiMsJiokLCZJInNHNiIiIiIhIiJGKCEiJEYoKiQsJkYmRihGKEYoRipGKQ== invlaplace(1/((s-1)^3)-1/((s+1)^3),s,t); NiMqJkkidEc2IiIiIy1JJXNpbmhHNiRJKnByb3RlY3RlZEdGKkkoX3N5c2xpYkdGJTYjRiQiIiI= laplace(Heaviside(t-3),t,s); NiMqJi1JJGV4cEc2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiMsJEkic0dGKSEiJCIiIkYsISIi invlaplace(exp(-3*s)/s,s,t); NiMtSSpIZWF2aXNpZGVHNiRJKnByb3RlY3RlZEdGJkkoX3N5c2xpYkc2IjYjLCZJInRHRigiIiIhIiRGLA== Discontinuous Functions Now lets look at some discontinuous functions f(t):=piecewise(0<=t and t<1, 1-t, 1<=t, 1); NiM+LUkiZkc2IjYjSSJ0R0YmLUkqUElFQ0VXSVNFR0YmNiQ3JCwmIiIiRi5GKCEiIjMxIiIhRigyRihGLjckRi4xRi5GKA== plot(f(t),t=0..5,discont=true); 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 laplace(f(t),t,s); NiMsJiokSSJzRzYiISIiIiIiKiZGJSEiIywmRidGKComLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRjBJKF9zeXNsaWJHRiY2IywkRiVGJ0YoLCZGJUYoRihGKEYoRihGKEYo This is not too good! so lets try to help Maple along. g(t):=(1-t)*(Heaviside(t)-Heaviside(t-1))+Heaviside(t-1); NiM+LUkiZ0c2IjYjSSJ0R0YmLCYqJiwmIiIiRixGKCEiIkYsLCYtSSpIZWF2aXNpZGVHNiRJKnByb3RlY3RlZEdGMkkoX3N5c2xpYkdGJkYnRiwtRjA2IywmRi1GLEYoRixGLUYsRixGNEYs plot(g(t),t=0..5,discont=true); 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laplace(g(t),t,s); NiMsJiokSSJzRzYiISIiIiIiKiZGJSEiIywmRidGKComLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRjBJKF9zeXNsaWJHRiY2IywkRiVGJ0YoLCZGJUYoRihGKEYoRihGKEYo h(t):=Heaviside(sin(2*t)); NiM+LUkiaEc2IjYjSSJ0R0YmLUkqSGVhdmlzaWRlRzYkSSpwcm90ZWN0ZWRHRixJKF9zeXNsaWJHRiY2Iy1JJHNpbkdGKzYjLCRGKCIiIw== plot(h(t),t=0..12,discont=true); 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laplace(h(t),t,s); NiMtSShsYXBsYWNlRzYiNiUtSSpIZWF2aXNpZGVHNiRJKnByb3RlY3RlZEdGKkkoX3N5c2xpYkdGJTYjLUkkc2luR0YpNiMsJEkidEdGJSIiI0YxSSJzR0Yl 1/(1-exp(-Pi*s))*int(exp(-s*t),t=0..Pi/2); NiMsJCooLCYiIiJGJi1JJGV4cEc2JEkqcHJvdGVjdGVkR0YqSShfc3lzbGliRzYiNiMsJComSSNQaUdGKkYmSSJzR0YsRiYhIiJGMkYyLCZGMkYmLUYoNiMsJEYvI0YyIiIjRiZGJkYxRjJGMg== invlaplace(-(exp(-1/2*Pi*s)-1)/((1-exp(-Pi*s))*s),s,t); NiMsJiMiIiIiIiNGJSkhIiItSSZmbG9vckc2JEkqcHJvdGVjdGVkR0YsSShfc3lzbGliRzYiNiMsJComSSJ0R0YuRiVJI1BpR0YsRihGJkYk Differential Equations Consider the differential equation NiMsJiooJSJkRyIiIyUieEciIiIqJCUjZHRHRiYhIiJGKEYmRig= NiMvLCYqJiUjZHhHIiIiJSNkdEchIiJGJyUieEdGJy0lImZHNiMlInRH NiMvLSUieEc2IyIiIUYn NiMqJiUjZHhHIiIiJSNkdEchIiI= (0)=0,where NiMvLSUiZkc2IyUidEctJSpQSUVDRVdJU0VHNiQ3JCIiIjMxLCRGJyEiIiIiITJGJ0YsNyRGMTFGLEYn . We define the right hand side, differential equation, and initial conditions. f(t):=Heaviside(t)-Heaviside(t-2); NiM+LUkiZkc2IjYjSSJ0R0YmLCYtSSpIZWF2aXNpZGVHNiRJKnByb3RlY3RlZEdGLUkoX3N5c2xpYkdGJkYnIiIiLUYrNiMsJkYoRi8hIiNGLyEiIg== de:=diff(x(t),t,t)+2*diff(x(t),t)+x(t)=f(t); NiM+SSNkZUc2Ii8sKC1JJWRpZmZHSSpwcm90ZWN0ZWRHRio2JC1JInhHRiU2I0kidEdGJS1JIiRHRio2JEYvIiIjIiIiLUYpNiRGLEYvRjNGLEY0LCYtSSpIZWF2aXNpZGVHNiRGKkkoX3N5c2xpYkdGJUYuRjQtRjk2IywmRi9GNCEiI0Y0ISIi ic:={x(0)=0,D(x)(0)=0}; NiM+SSNpY0c2IjwkLy1JInhHRiU2IyIiIUYrLy0tSSJERzYkSSpwcm90ZWN0ZWRHRjFJKF9zeXNsaWJHRiU2I0YpRipGKw== Laplace transform the differential equation. LEq:=laplace(de,t,s); NiM+SSRMRXFHNiIvLC4qJkkic0dGJSIiIy1JKGxhcGxhY2VHRiU2JS1JInhHRiU2I0kidEdGJUYxRikiIiJGMi0tSSJERzYkSSpwcm90ZWN0ZWRHRjdJKF9zeXNsaWJHRiU2I0YvNiMiIiEhIiIqJkYpRjItRi9GOkYyRjwqJkYpRjJGK0YyRipGPiEiI0YrRjIqJkYpRjwsJkYyRjItSSRleHBHRjY2IywkRilGQEY8RjI= Substitute the values for the initial conditions and solve the resulting algebraic equation for the transform of the solution. LEqIc:=subs(ic,LEq); NiM+SSZMRXFJY0c2Ii8sKComSSJzR0YlIiIjLUkobGFwbGFjZUdGJTYlLUkieEdGJTYjSSJ0R0YlRjFGKSIiIkYyKiZGKUYyRitGMkYqRitGMiomRikhIiIsJkYyRjItSSRleHBHNiRJKnByb3RlY3RlZEdGOkkoX3N5c2xpYkdGJTYjLCRGKSEiI0Y1RjI= LT:=solve(LEqIc,laplace(x(t),t,s)); NiM+SSNMVEc2IiwkKigsJiEiIiIiIi1JJGV4cEc2JEkqcHJvdGVjdGVkR0YuSShfc3lzbGliR0YlNiMsJEkic0dGJSEiI0YqRipGMkYpLCgqJEYyIiIjRipGMkY2RipGKkYpRik= Invert to find the solution. invlaplace(LT,s,t); NiMsKCIiIkYkKiYtSSRleHBHNiRJKnByb3RlY3RlZEdGKUkoX3N5c2xpYkc2IjYjLCRJInRHRishIiJGJCwmRi5GJEYkRiRGJEYvKiYtSSpIZWF2aXNpZGVHRig2IywmRi5GJCEiI0YkRiQsJkYvRiQqJi1GJzYjLCYiIiNGJEYuRi9GJCwmRi9GJEYuRiRGJEYkRiRGJA== sol:=%; NiM+SSRzb2xHNiIsKCIiIkYnKiYtSSRleHBHNiRJKnByb3RlY3RlZEdGLEkoX3N5c2xpYkdGJTYjLCRJInRHRiUhIiJGJywmRjBGJ0YnRidGJ0YxKiYtSSpIZWF2aXNpZGVHRis2IywmRjBGJyEiI0YnRicsJkYxRicqJi1GKjYjLCYiIiNGJ0YwRjFGJywmRjFGJ0YwRidGJ0YnRidGJw== And finally graph the solution. plot(sol,t=0..8); 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 Consider the differential equation NiMqKCUiZEciIiMlInhHIiIiKiQlI2R0R0YlISIi + NiMvKiYiIiUiIiIlInhHRiYtJSJmRzYjJSJ0Rw== NiMvLSUieEc2IyIiISomIiIiRikiIiMhIiI= NiMqJiUjZHhHIiIiJSNkdEchIiI= (0)=0,where NiMvLSUiZkc2IyUidEctJSpQSUVDRVdJU0VHNiQ3JCwmIiIlIiIiKiYiIiNGLkYnRi4hIiIzMSwkRidGMSIiITJGJ0YtNyRGNTFGLUYn . We define the right hand side, differential equation, and initial conditions. f(t) := piecewise(0 <= t and t < 4,4-2*t,4 <= t,0); NiM+LUkiZkc2IjYjSSJ0R0YmLUkqUElFQ0VXSVNFR0YmNiQ3JCwmIiIlIiIiRighIiMzMSIiIUYoMkYoRi43JEYzMUYuRig= Note: Maple can convert automatically to notation using the Heaviside function. f(t):=convert(%,Heaviside); NiM+LUkiZkc2IjYjSSJ0R0YmLCotSSpIZWF2aXNpZGVHNiRJKnByb3RlY3RlZEdGLUkoX3N5c2xpYkdGJkYnIiIlLUYrNiMsJiEiJSIiIkYoRjRGMyomRihGNEYqRjQhIiMqJkYoRjRGMEY0IiIj de:=diff(x(t),t,t)+4*x(t)=f(t); NiM+SSNkZUc2Ii8sJi1JJWRpZmZHSSpwcm90ZWN0ZWRHRio2JC1JInhHRiU2I0kidEdGJS1JIiRHRio2JEYvIiIjIiIiRiwiIiUsKi1JKkhlYXZpc2lkZUc2JEYqSShfc3lzbGliR0YlRi5GNS1GODYjLCYhIiVGNEYvRjRGPiomRi9GNEY3RjQhIiMqJkYvRjRGO0Y0RjM= ic:={x(0)=1/2,D(x)(0)=0}; NiM+SSNpY0c2IjwkLy1JInhHRiU2IyIiISMiIiIiIiMvLS1JIkRHNiRJKnByb3RlY3RlZEdGNEkoX3N5c2xpYkdGJTYjRilGKkYr Laplace transform the differential equation. LEq:=laplace(de,t,s); NiM+SSRMRXFHNiIvLCoqJkkic0dGJSIiIy1JKGxhcGxhY2VHRiU2JS1JInhHRiU2I0kidEdGJUYxRikiIiJGMi0tSSJERzYkSSpwcm90ZWN0ZWRHRjdJKF9zeXNsaWJHRiU2I0YvNiMiIiEhIiIqJkYpRjItRi9GOkYyRjxGKyIiJSwmKiRGKUY8Rj8qJkYpISIjLCZGPEYyKiYsJkYpRipGMkYyRjItSSRleHBHRjY2IywkRikhIiVGMkYyRjJGKg== Substitute the values for the initial conditions and solve the resulting algebraic equation for the transform of the solution. LEqIc:=subs(ic,LEq); NiM+SSZMRXFJY0c2Ii8sKComSSJzR0YlIiIjLUkobGFwbGFjZUdGJTYlLUkieEdGJTYjSSJ0R0YlRjFGKSIiIkYyRikjISIiRipGKyIiJSwmKiRGKUY0RjUqJkYpISIjLCZGNEYyKiYsJkYpRipGMkYyRjItSSRleHBHNiRJKnByb3RlY3RlZEdGQEkoX3N5c2xpYkdGJTYjLCRGKSEiJUYyRjJGMkYq LT:=solve(LEqIc,laplace(x(t),t,s)); NiM+SSNMVEc2IiwkKigsLCokSSJzR0YlIiIkIiIiRioiIikhIiVGLComLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRjNJKF9zeXNsaWJHRiU2IywkRipGLkYsRipGLEYtRjAiIiVGLEYqISIjLCYqJEYqIiIjRixGN0YsISIiI0YsRjs= Invert to find the solution. invlaplace(LT,s,t); NiMsLCIiIkYkLUkkY29zRzYkSSpwcm90ZWN0ZWRHRihJKF9zeXNsaWJHNiI2IywkSSJ0R0YqIiIjIyEiIkYuRi1GLy1JJHNpbkdGJ0YrI0YkIiIlKiYtSSpIZWF2aXNpZGVHRic2IywmISIlRiRGLUYkRiQsKiEiKUYkKiQtRjJGOEYuIiIpLUYyNiMsJkY8RiRGLUYuRjBGLUYuRiRGMw== sol:=%; NiM+SSRzb2xHNiIsLCIiIkYnLUkkY29zRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiU2IywkSSJ0R0YlIiIjIyEiIkYwRi9GMS1JJHNpbkdGKkYtI0YnIiIlKiYtSSpIZWF2aXNpZGVHRio2IywmISIlRidGL0YnRicsKiEiKUYnKiQtRjRGOkYwIiIpLUY0NiMsJkY+RidGL0YwRjJGL0YwRidGNQ== And finally graph the solution. plot(sol,t=0..9); 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