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. Problem 17 Consider the following problem (problem 17 on page 76): A patient receives a drug intravenously for a kidney infection. The drug is infused into the patient's blood at the rate of 0.10 cc/min. After 2 hours from the onset of the treatment, the drug is absorbed by the kidneys at a rate of 5% of the amount in the blood at that time. Formulate an i.v.p. for the amount of the drug present in the blood at any given time NiMlInRH . Let NiMlInhH (NiMlInRH ) be the amount of drug in the blood at any given time NiMlInRH . Then: NiMqJiUjZHhHIiIiJSNkdEchIiI= =NiMtJSpQSUVDRVdJU0VHNiQ3JCQiIiIhIiIzMSIiISUidEcxRi0iJD8iNyQsJkYnRigqJiQiIiYhIiNGKCUieEdGKEYpMkYvRi0= x(0)=0 restart:with(DEtools):with(plots): Warning, the name changecoords has been redefined We can try to look at the direction field...(note the piecewise definition) DEplot(D(x)(t)=piecewise(0<=t and t<=120,0.1, 120 < t,0.1-0.05*x(t)),x(t),t=0..200,x=0..3); 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In order to solve the problem, we can split the problem into two problems NiMvKiYmJSNkeEc2IyIiIkYoJSNkdEchIiIkRihGKg== for NiMxIiIhJSJ0Rw== and NiMxJSJ0RyIkPyI= NiMvLSYlInhHNiMiIiI2IyIiIUYq and NiMvKiYmJSNkeEc2IyIiIyIiIiUjZHRHISIiLCYkRilGK0YpKiYkIiImISIjRikmJSJ4R0YnRilGKw== for NiMyIiQ/IiUidEc= NiMvLSYlInhHNiMiIiM2IyIkPyItJkYmNiMiIiJGKQ== We now write the equations and solve them. eq1:=diff(x[1](t),t)=0.1; NiM+SSRlcTFHNiIvLUklZGlmZkdJKnByb3RlY3RlZEdGKTYkLSZJInhHRiU2IyIiIjYjSSJ0R0YlRjEkRi8hIiI= ic1:=x[1](0)=0; NiM+SSRpYzFHNiIvLSZJInhHRiU2IyIiIjYjIiIhRi0= eq2:=diff(x[2](t),t)=0.1-0.05*x[2](t); NiM+SSRlcTJHNiIvLUklZGlmZkdJKnByb3RlY3RlZEdGKTYkLSZJInhHRiU2IyIiIzYjSSJ0R0YlRjEsJiQiIiIhIiJGNEYrJCEiJiEiIw== ic2:=x[2](120)=x[1](120); NiM+SSRpYzJHNiIvLSZJInhHRiU2IyIiIzYjIiQ/Ii0mRik2IyIiIkYs sol1:=rhs(dsolve({eq1,ic1},x[1](t))); NiM+SSVzb2wxRzYiLCRJInRHRiUjIiIiIiM1 x[1](120):=subs(t=120,sol1); NiM+LSZJInhHNiI2IyIiIjYjIiQ/IiIjNw== sol2:=rhs(dsolve({eq2,ic2},x[2](t))); NiM+SSVzb2wyRzYiLCYiIiMiIiIqJi1JJGV4cEc2JEkqcHJvdGVjdGVkR0YtSShfc3lzbGliR0YlNiMsJEkidEdGJSMhIiIiIz9GKC1GKzYjISInRjMiIzU= sol:=piecewise(0<=t and t<=120,sol1,120<t,sol2); NiM+SSRzb2xHNiItSSpQSUVDRVdJU0VHRiU2JDckLCRJInRHRiUjIiIiIiM1MzEiIiFGKzFGKyIkPyI3JCwmIiIjRi0qJi1JJGV4cEc2JEkqcHJvdGVjdGVkR0Y7SShfc3lzbGliR0YlNiMsJEYrIyEiIiIjP0YtLUY5NiMhIidGQEYuMkYzRis= plot(sol,t=0..150); 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 Problem 18 Consider the following problem (problem 18 on page 76): A drug is administered intravenously to a patient at a rate of 1,000 units/ml/s. The drug is absorbed immediately into the patient's bloodstream and metabolized by the body (i.e., removed from the bloodstream) at a rate of 1% of the amount of the drug present at the time. The intravenous process is stopped after 2 minutes. Formulate an i.v.p. for the amount of drug present in the blood at any time NiMlInRH . Then: NiMqJiUjZHhHIiIiJSNkdEchIiI= =NiMtJSpQSUVDRVdJU0VHNiQ3JCwmIiUrNSIiIiomJEYpISIjRiklInhHRikhIiIzMSIiISUidEcxRjIiJD8iNyQsJEYqRi4yRjRGMg== x(0)=0 We can try to look at the direction field...(note the piecewise definition) DEplot(D(x)(t)=piecewise(0<=t and t<=122,1000-0.01*x(t), 120 < t,-0.01*x(t)),x(t),t=0..300,x=0..150000); 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eq1:=diff(x[1](t),t)=1000-0.01*x[1](t); NiM+SSRlcTFHNiIvLUklZGlmZkdJKnByb3RlY3RlZEdGKTYkLSZJInhHRiU2IyIiIjYjSSJ0R0YlRjEsJiIlKzVGL0YrJCEiIiEiIw== ic1:=x[1](0)=0; NiM+SSRpYzFHNiIvLSZJInhHRiU2IyIiIjYjIiIhRi0= eq2:=diff(x[2](t),t)=-0.01*x[2](t); NiM+SSRlcTJHNiIvLUklZGlmZkdJKnByb3RlY3RlZEdGKTYkLSZJInhHRiU2IyIiIzYjSSJ0R0YlRjEsJEYrJCEiIiEiIw== x[1](120):='x[1](120)';ic2:=x[2](120)=x[1](120); NiM+LSZJInhHNiI2IyIiIjYjIiQ/IkYk NiM+SSRpYzJHNiIvLSZJInhHRiU2IyIiIzYjIiQ/Ii0mRik2IyIiIkYs sol1:=rhs(dsolve({eq1,ic1},x[1](t))); NiM+SSVzb2wxRzYiLCYiJysrNSIiIi1JJGV4cEc2JEkqcHJvdGVjdGVkR0YsSShfc3lzbGliR0YlNiMsJEkidEdGJSMhIiIiJCsiIScrKzU= x[1](120):=subs(t=120,sol1); NiM+LSZJInhHNiI2IyIiIjYjIiQ/IiwmIicrKzVGKS1JJGV4cEc2JEkqcHJvdGVjdGVkR0YxSShfc3lzbGliR0YnNiMjISInIiImIScrKzU= sol2:=rhs(dsolve({eq2,ic2},x[2](t))); NiM+SSVzb2wyRzYiLCQqKC1JJGV4cEc2JEkqcHJvdGVjdGVkR0YrSShfc3lzbGliR0YlNiMsJEkidEdGJSMhIiIiJCsiIiIiLCYhJysrNUYzLUYpNiMjISInIiImIicrKzVGM0Y2RjFGMQ== sol:=piecewise(0<=t and t<=120,sol1,120<t,sol2); NiM+SSRzb2xHNiItSSpQSUVDRVdJU0VHRiU2JDckLCYiJysrNSIiIi1JJGV4cEc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiMsJEkidEdGJSMhIiIiJCsiIScrKzUzMSIiIUY0MUY0IiQ/IjckLCQqKEYtRiwsJkY4RiwtRi42IyMhIiciIiZGK0YsRkJGNkY2MkY9RjQ= plot(sol,t=0..300); 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