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: 10


The actual result: $4$ The approximation result: $4.8$ Difference: $4.8-4=0.8$

Work Step by Step

We are given $\frac{dy}{dx}=4x+3$ Using Euler's method $g(x,y)=4x+3$ Since $x=1, y=0$ $g(x_{0},y_{0})=7$ and $y_{1}=y_{0}+g(x_{0},y_{0})h=0+7\times0.1=0.7$ Now $x_{1}=1.1, y_{1}=0.7$ and $g(x_{1},y_{1})=7.4$ Then $y_{2}=0.7+(7.4)\times0.1=1.44$ $y_{3}=1.44+(7.8)\times0.1=2.22$ $y_{4}=2.22+(8.2)\times0.1=3.04$ $y_{5}=3.04+(8.6)\times0.1=3.9$ $y_{6}=3.9+(9)\times0.1=4.8$ $y(1.5)=4.8$ $\frac{dy}{dx}=4x+3$ $\int dy=\int (4x+3)dx$ $y=2x^{2} +3x+C$ With $x=1, y=0$ $C=0-2(1)^{2}-3(1)=-5$ $y=2x^{2} +3x-5$ $y(1.5)=2(1.5)^{2}+3(1.5)-5=4$ Difference between the result and approximation: $4.8-4=0.8$
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