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. We will now look at (linearity and at) linear second order differential equations. restart:with(DEtools):with(plots): Warning, the name changecoords has been redefined Consider the equation NiMsJiooJSJkRyIiIyUieEciIiIqJCUjZHRHRiYhIiJGKC0lImFHNiMlInRHRig= NiMsJiomJSNkeEciIiIlI2R0RyEiIkYmLSUiYkc2IyUidEdGJg== NiMvJSJ4Ry0lImdHNiMlInRH NiMvLSUieEc2IyIiISZGJUYm , NiMqJiUjZHhHIiIiJSNkdEchIiI= (0)=NiMmJSJ2RzYjIiIh .This is the prototype linear, second order, differential equation and initial conditions. Together they form an initial value problem (i.v.p.). We consider the differential equation (d.e.): (*) NiMsJiooJSJkRyIiIyUieEciIiIqJCUjZHRHRiYhIiJGKC0lImFHNiMlInRHRig= NiMsJiomJSNkeEciIiIlI2R0RyEiIkYmLSUiYkc2IyUidEdGJg== NiMvJSJ4Ry0lImdHNiMlInRH and the corresponding homogeneous d.e. (**) NiMsJiooJSJkRyIiIyUieEciIiIqJCUjZHRHRiYhIiJGKC0lImFHNiMlInRHRig= NiMsJiomJSNkeEciIiIlI2R0RyEiIkYmLSUiYkc2IyUidEdGJg== NiMvJSJ4RyIiIQ== (note the zero r.h.s.). Due to linearity if NiMmJSRwaGlHNiMiIiI= and NiMmJSRwaGlHNiMiIiM= are two solutions of the homogeneous d.e. (**) then also so is every linear combination of NiMmJSRwaGlHNiMiIiI= and NiMmJSRwaGlHNiMiIiM= , i.e., NiMvLSUkcGhpRzYjJSJ0RywmKiYmJSJjRzYjIiIiRi0tJkYlRixGJkYtRi0qJiZGKzYjIiIjRi0tJkYlRjJGJkYtRi0= , where NiMmJSJjRzYjIiIi and NiMmJSJjRzYjIiIj are arbitrary constants. That is for any constants NiMmJSJjRzYjIiIi and NiMmJSJjRzYjIiIj , NiMvLSUkcGhpRzYjJSJ0RywmKiYmJSJjRzYjIiIiRi0tJkYlRixGJkYtRi0qJiZGKzYjIiIjRi0tJkYlRjJGJkYtRi0= is a solution of (**). Also assume that NiMmJSRwaGlHNiMlImdH is a solution of (*), (a particular solution), then due to linearity, for any constants NiMmJSJjRzYjIiIi and NiMmJSJjRzYjIiIj , NiMvLSUkcGhpRzYjJSJ0RywoLSZGJTYjJSJnR0YmIiIiKiYmJSJjRzYjRi1GLS0mRiVGMUYmRi1GLSomJkYwNiMiIiNGLS0mRiVGNkYmRi1GLQ== is a solution of (*). Furthermore if NiMmJSRwaGlHNiMlImZH is a solution of (*) with the r.h.s. NiMtJSJnRzYjJSJ0Rw== replaced by NiMtJSJmRzYjJSJ0Rw== , then (due to linearity) NiMvLSUkcGhpRzYjJSJ0RywmLSZGJTYjJSJnR0YmIiIiLSZGJTYjJSJmR0YmRi0= is a solution of (*) with the r.h.s. NiMtJSJnRzYjJSJ0Rw== replaced by NiMsJi0lImdHNiMlInRHIiIiLSUiZkdGJkYo and also for any constants NiMmJSJjRzYjIiIi and NiMmJSJjRzYjIiIj , NiMvLSUkcGhpRzYjJSJ0RywqLSZGJTYjJSJnR0YmIiIiLSZGJTYjJSJmR0YmRi0qJiYlImNHNiNGLUYtLSZGJUY1RiZGLUYtKiYmRjQ2IyIiI0YtLSZGJUY6RiZGLUYt is a solution of (*) with the r.h.s. NiMtJSJnRzYjJSJ0Rw== replaced by NiMsJi0lImdHNiMlInRHIiIiLSUiZkdGJkYo . Examples: Lets define the differential equation: de:=D(D(x))(t)+a*D(x)(t)+b*x(t)=g(t); NiM+SSNkZUc2Ii8sKC0tLUkjQEBHNiRJKnByb3RlY3RlZEdGLUkoX3N5c2xpYkdGJTYkSSJER0YsIiIjNiNJInhHRiU2I0kidEdGJSIiIiomSSJhR0YlRjYtLUYwRjJGNEY2RjYqJkkiYkdGJUY2LUYzRjRGNkY2LUkiZ0dGJUY0 ic:=x(0)=x_0,D(x)(0)=v_0; NiM+SSNpY0c2IjYkLy1JInhHRiU2IyIiIUkkeF8wR0YlLy0tSSJERzYkSSpwcm90ZWN0ZWRHRjJJKF9zeXNsaWJHRiU2I0YpRipJJHZfMEdGJQ== We will now consider homogeneous equations (with NiMvLSUiZ0c2IyUidEciIiE= ) and with constant coefficients (NiMlImFH and NiMlImJH constants). We will choose some values for NiMlImFH and NiMlImJH and solve the equation using the dsolve command. You can also try solving the equation by hand. g(t):=0;a:=0;b:=9; NiM+LUkiZ0c2IjYjSSJ0R0YmIiIh NiM+SSJhRzYiIiIh NiM+SSJiRzYiIiIq sol:=dsolve(de,x(t)); NiM+SSRzb2xHNiIvLUkieEdGJTYjSSJ0R0YlLCYqJkkkX0MxR0YlIiIiLUkkc2luRzYkSSpwcm90ZWN0ZWRHRjJJKF9zeXNsaWJHRiU2IywkRioiIiRGLkYuKiZJJF9DMkdGJUYuLUkkY29zR0YxRjRGLkYu g(t):=cos(t); NiM+LUkiZ0c2IjYjSSJ0R0YmLUkkY29zRzYkSSpwcm90ZWN0ZWRHRixJKF9zeXNsaWJHRiZGJw== sol:=dsolve(de,x(t)); NiM+SSRzb2xHNiIvLUkieEdGJTYjSSJ0R0YlLCgqJi1JJHNpbkc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiMsJEYqIiIkIiIiSSRfQzJHRiVGNUY1KiYtSSRjb3NHRi9GMkY1SSRfQzFHRiVGNUY1LUY5RikjRjUiIik= Observe the homogeneous solution is NiM+JiUieEc2IyUiaEcsJiomJiUiY0c2IyIiIkYtLSUkc2luRzYjKiYiIiRGLSUidEdGLUYtRi0qJiZGKzYjIiIjRi0tJSRjb3NHRjBGLUYt and the particular solution is NiMvJiUieEc2IyUicEcqJi0lJGNvc0c2IyUidEciIiIiIikhIiI= . g(t):=0; NiM+LUkiZ0c2IjYjSSJ0R0YmIiIh x_0:=1;v_0:=0; NiM+SSR4XzBHNiIiIiI= NiM+SSR2XzBHNiIiIiE= de;ic; NiMvLCYtLS1JI0BARzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHNiI2JEkiREdGKSIiIzYjSSJ4R0YsNiNJInRHRiwiIiItRjFGMiIiKiIiIQ== NiQvLUkieEc2IjYjIiIhIiIiLy0tSSJERzYkSSpwcm90ZWN0ZWRHRi9JKF9zeXNsaWJHRiY2I0YlRidGKA== sol1:=dsolve({de,ic},x(t)); NiM+SSVzb2wxRzYiLy1JInhHRiU2I0kidEdGJS1JJGNvc0c2JEkqcHJvdGVjdGVkR0YuSShfc3lzbGliR0YlNiMsJEYqIiIk sol1:=rhs(sol1); NiM+SSVzb2wxRzYiLUkkY29zRzYkSSpwcm90ZWN0ZWRHRilJKF9zeXNsaWJHRiU2IywkSSJ0R0YlIiIk plot(sol1,t=0..4); 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 g(t):=sin(2*t); NiM+LUkiZ0c2IjYjSSJ0R0YmLUkkc2luRzYkSSpwcm90ZWN0ZWRHRixJKF9zeXNsaWJHRiY2IywkRigiIiM= de;ic; NiMvLCYtLS1JI0BARzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHNiI2JEkiREdGKSIiIzYjSSJ4R0YsNiNJInRHRiwiIiItRjFGMiIiKi1JJHNpbkdGKTYjLCRGM0Yv NiQvLUkieEc2IjYjIiIhIiIiLy0tSSJERzYkSSpwcm90ZWN0ZWRHRi9JKF9zeXNsaWJHRiY2I0YlRidGKA== sol2:=dsolve({de,ic},x(t)); NiM+SSVzb2wyRzYiLy1JInhHRiU2I0kidEdGJSwoLUkkc2luRzYkSSpwcm90ZWN0ZWRHRi9JKF9zeXNsaWJHRiU2IywkRioiIiQjISIjIiM6LUkkY29zR0YuRjEiIiItRi02IywkRioiIiMjRjkiIiY= sol2r:=rhs(sol2); NiM+SSZzb2wyckc2IiwoLUkkc2luRzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHRiU2IywkSSJ0R0YlIiIkIyEiIyIjOi1JJGNvc0dGKUYsIiIiLUYoNiMsJEYuIiIjI0Y1IiIm plot(sol2r,t=0..13); 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xh:=select(has,sol2r,3*t); NiM+SSN4aEc2IiwmLUkkc2luRzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHRiU2IywkSSJ0R0YlIiIkIyEiIyIjOi1JJGNvc0dGKUYsIiIi xp:=select(has,sol2r,2*t); NiM+SSN4cEc2IiwkLUkkc2luRzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHRiU2IywkSSJ0R0YlIiIjIyIiIiIiJg== odetest(x(t)=cos(3*t)-2/15*sin(3*t),de); NiMsJC1JJHNpbkc2JEkqcHJvdGVjdGVkR0YnSShfc3lzbGliRzYiNiMsJEkidEdGKSIiIyEiIg== odetest(x(t)=1/5*sin(2*t),de); NiMiIiE= g(t):=sin(3*t); NiM+LUkiZ0c2IjYjSSJ0R0YmLUkkc2luRzYkSSpwcm90ZWN0ZWRHRixJKF9zeXNsaWJHRiY2IywkRigiIiQ= x_0:=1;v_0:=0; NiM+SSR4XzBHNiIiIiI= NiM+SSR2XzBHNiIiIiE= de;ic; NiMvLCYtLS1JI0BARzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHNiI2JEkiREdGKSIiIzYjSSJ4R0YsNiNJInRHRiwiIiItRjFGMiIiKi1JJHNpbkdGKTYjLCRGMyIiJA== NiQvLUkieEc2IjYjIiIhIiIiLy0tSSJERzYkSSpwcm90ZWN0ZWRHRi9JKF9zeXNsaWJHRiY2I0YlRidGKA== sol3:=dsolve({de,ic},x(t)); NiM+SSVzb2wzRzYiLy1JInhHRiU2I0kidEdGJSwoLUkkc2luRzYkSSpwcm90ZWN0ZWRHRi9JKF9zeXNsaWJHRiU2IywkRioiIiQjIiIiIiM9LUkkY29zR0YuRjFGNSomRjdGNUYqRjUjISIiIiIn sol3r:=rhs(sol3); NiM+SSZzb2wzckc2IiwoLUkkc2luRzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHRiU2IywkSSJ0R0YlIiIkIyIiIiIjPS1JJGNvc0dGKUYsRjEqJkYzRjFGLkYxIyEiIiIiJw== plot(sol3r,t=0..13); 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g(t):=sin(2*t); NiM+LUkiZ0c2IjYjSSJ0R0YmLUkkc2luRzYkSSpwcm90ZWN0ZWRHRixJKF9zeXNsaWJHRiY2IywkRigiIiM= sol4:=dsolve(de,x(t)); NiM+SSVzb2w0RzYiLy1JInhHRiU2I0kidEdGJSwoKiYtSSRzaW5HNiRJKnByb3RlY3RlZEdGMEkoX3N5c2xpYkdGJTYjLCRGKiIiJCIiIkkkX0MyR0YlRjVGNSomLUkkY29zR0YvRjJGNUkkX0MxR0YlRjVGNS1GLjYjLCRGKiIiIyNGNSIiJg== sol4:=rhs(sol4); NiM+SSVzb2w0RzYiLCgqJi1JJHNpbkc2JEkqcHJvdGVjdGVkR0YrSShfc3lzbGliR0YlNiMsJEkidEdGJSIiJCIiIkkkX0MyR0YlRjFGMSomLUkkY29zR0YqRi1GMUkkX0MxR0YlRjFGMS1GKTYjLCRGLyIiIyNGMSIiJg== xh:=select(has,sol4,3*t); NiM+SSN4aEc2IiwmKiYtSSRzaW5HNiRJKnByb3RlY3RlZEdGK0koX3N5c2xpYkdGJTYjLCRJInRHRiUiIiQiIiJJJF9DMkdGJUYxRjEqJi1JJGNvc0dGKkYtRjFJJF9DMUdGJUYxRjE= xp:=select(has,sol4,2*t); NiM+SSN4cEc2IiwkLUkkc2luRzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHRiU2IywkSSJ0R0YlIiIjIyIiIiIiJg== x_0:='x_0';v_0:='v_0'; NiM+SSR4XzBHNiJGJA== NiM+SSR2XzBHNiJGJA== eq1:=subs(t=0,xh+xp)=x_0; NiM+SSRlcTFHNiIvLCgqJi1JJHNpbkc2JEkqcHJvdGVjdGVkR0YsSShfc3lzbGliR0YlNiMiIiEiIiJJJF9DMkdGJUYwRjAqJi1JJGNvc0dGK0YuRjBJJF9DMUdGJUYwRjBGKSNGMCIiJkkkeF8wR0Yl eq2:=subs(t=0,diff((xh+xp),t))=v_0; NiM+SSRlcTJHNiIvLCgqJkkkX0MyR0YlIiIiLUkkY29zRzYkSSpwcm90ZWN0ZWRHRi5JKF9zeXNsaWJHRiU2IyIiIUYqIiIkKiZJJF9DMUdGJUYqLUkkc2luR0YtRjBGKiEiJEYrIyIiIyIiJkkkdl8wR0Yl eval(eq1); NiMvSSRfQzFHNiJJJHhfMEdGJQ== eval(eq2); NiMvLCZJJF9DMkc2IiIiJCMiIiMiIiYiIiJJJHZfMEdGJg== Exercise: Solve the equation: NiMvLCYqKCUiZEciIiMlInhHIiIiKiQlI2R0R0YnISIiRikqJiIjO0YpRihGKUYpLSUkU2luRzYjKiZGJ0YpJSJ0R0Yp NiMvLSUieEc2IyIiIUYn , NiMqJiUjZHhHIiIiJSNkdEchIiI= (0)=NiMiIiI= .Identify the homogeneous solution and the particular solution. Is the solution periodic, if it is what is the period? Repeat the above for NiMvLCYqKCUiZEciIiMlInhHIiIiKiQlI2R0R0YnISIiRikqJiIjO0YpRihGKUYpLSUkU2luRzYjKiYiIiVGKSUidEdGKQ== NiMvLSUieEc2IyIiIUYn , NiMqJiUjZHhHIiIiJSNkdEchIiI= (0)=NiMiIiI= .