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Advanced Engineering Mathematics, International Student Edition

Peter V. O'Neil

Chapter 5

Numerical Approximation of Solutions - all with Video Answers

Educators

Section 1

Euler's Method

Problem 1

Obtain the exact solution and graph it with the approximate solutions.
$y^{\prime}=y \sin (x) ; y(0)=1$

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04:00

Problem 2

Obtain the exact solution and graph it with the approximate solutions.
$y^{\prime}=x+y: y(1)=-3$

Linda Winkler
Linda Winkler
Numerade Educator
05:58

Problem 3

Obtain the exact solution and graph it with the approximate solutions.
$y^{\prime}=3 x y ; y(0)=5$

Mike Gaerlan
Mike Gaerlan
Numerade Educator
04:04

Problem 4

Obtain the exact solution and graph it with the approximate solutions.
$y^{\prime}=2-x ; y(0)=1$

Anthony Ramos
Anthony Ramos
Numerade Educator
01:02

Problem 5

Obtain the exact solution and graph it with the approximate solutions.
$y^{\prime}=y-\cos (x) ; y(1)=-2$

Clarissa Noh
Clarissa Noh
Numerade Educator
02:53

Problem 6

Obtain the exact solution and graph it with the approximate solutions.
$y^{\prime}=x-y^2 ; y(0)=4$

Vikash Ranjan
Vikash Ranjan
Numerade Educator

Problem 7

Approximate $e$ as follows. Use Euler's method with $h=0.01$ to approximate $y(1)$, where $y(x)$ is the solution of $y^{\prime}=y ; y(0)=1$. Sketch a graph of the solution before applying Euler's method and determine whether the approximate value obtained is less than or greater than the actual value.

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Problem 8

Approximate $\ln (2)$ by using Euler's method to approximate $y(2)$, where $y(x)$ is the solution of $y^{\prime}=1 / x ; y(1)=0$. Use $h=0.01$. Will this approximation be less than or greater than the actual value?

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Problem 9

In the analysis of the radicactive waste disposal problem, how does the constant of proportionality for the drag on the drum affect the conclusion? Carry out the numerical analysis if the drag is $0.3 e^{\sqrt{T / 3}}$, and again for the case that the drag is $0.8 \mathrm{v}^{\sqrt{70 / 3}}$.

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02:05

Problem 10

Try exponents other than $\sqrt{10} / 3$ for the velocity in the disposal problem to gauge the effect of this number on the conclusion. In particular, perform the analysis if the drag equals $0.5 v$ ( 1 is slightly less than $\sqrt{10} / 3$ ) and again for a drag effect of $0.5 v^{4 / 3}(4 / 3$ is slightly greater than $\sqrt{10} / 3$ ).

Manish Jain
Manish Jain
Numerade Educator
06:39

Problem 11

Suppose the drums are dropped over a part of the ocean having a depth of 340 feet. Will the drums be likely to rupture on impact with the ocean floor?

Hubert Agamasu
Hubert Agamasu
Numerade Educator