Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 10 - Differential Equations - 10.3 Euler's Method - 10.3 Exercises - Page 550: 15


The actual result $3.271$ The approximation: $3.574$ Difference: $3.574-3.271=0.303$

Work Step by Step

We are given $\frac{dy}{dx}=ye^{x}$ Using Euler's method $g(x,y)=ye^{x}$ Since $x=0, y=2$ $g(x_{0},y_{0})=2$ and $y_{1}=y_{0}+g(x_{0},y_{0})h=2+2\times0.1=2.2$ Now $x_{1}=0.1, y_{1}=2.2$ and $g(x_{1},y_{1})=2.431$ Then $y_{2}=2.2+(2.431)\times0.1=2.4431$ $y_{3}=2.4431+(2.98)\times0.1=2.741$ $y_{4}=2.741+(3.7)\times0.1=3.11$ $y_{5}=3.11+(4.641)\times0.1=3.574$ $y(0.4)=3.574$ $\frac{dy}{dx}=ye^{x}$ $\int\frac{1}{y}dy=\int e^{x}dx$ $\ln|y|=e^{x} + C$ With $x=0, y=2$ $C=-0.307$ $\ln|y|=e^{x} - 0.307$ $y(0.4) \rightarrow \ln|y|=e^{0.4} -0.307=1.185$ $\rightarrow y=e^{1.185}=3.271$ Difference between the result and approximation: $3.574-3.271=0.303$
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