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: 12

Answer

The actual result: $3.267$ The approximation: $3.009$ Difference: $3.267-3.009=0.258$

Work Step by Step

We are given $\frac{dy}{dx}=\frac{3}{x}$ Using Euler's method $g(x,y)=\frac{3}{x}$ Since $x=1, y=2$ $g(x_{0},y_{0})=3$ and $y_{1}=y_{0}+g(x_{0},y_{0})h=2+3\times0.1=2.3$ Now $x_{1}=1.1, y_{1}=2.3$ and $g(x_{1},y_{1})=2.727$ Then $y_{2}=2.3+(2.727)\times0.1=2.572$ $y_{3}=2.572+(2.5)\times0.1=2.822$ $y_{4}=2.822+(2.31)\times0.1=3.053$ $y_{5}=3.053+(2.143)\times0.1=3.267$ $y(1.4)=y_{7}=3.267$ $\frac{dy}{dx}=\frac{3}{x}$ $\int dy=\int \frac{3}{x}dx$ $y=3\ln|x|+C$ With $x=1, y=2$ $C=y-3 \ln1=2$ $y=3\ln|x|+2$ $y(1.4)=3\ln|1.4|+2=3.009$ Difference between the result and approximation: $3.267-3.009=0.258$
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