Given that y(x) is a solution to yx² - 2y + 1 and y(0) = 1, a. Find the first four terms in the Maclaurin series for y(x). (Hint: Write y(x) = ao +...+ a3x³ and plug this expression into the differential equation. Then you may equate coefficients of x on both sides of the equation.) b. Estimate the error when the polynomial you found in (a) is used to estimate y(0.1). You may use c = 0 in the remainder formula.
Given that y(x) is a solution to yx² - 2y + 1 and y(0) = 1, a. Find the first four terms in the Maclaurin series for y(x). (Hint: Write y(x) = ao +...+ a3x³ and plug this expression into the differential equation. Then you may equate coefficients of x on both sides of the equation.) b. Estimate the error when the polynomial you found in (a) is used to estimate y(0.1). You may use c = 0 in the remainder formula.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please do part A and B and please show steps and explain
Transcribed Image Text:These exercises emphasize two techniques: (a) Substitution; (b) term-by-term operation on series. These
techniques simplify calculations enormously. You should make use of these techniques in each
problem. The point of these exercises is to get you comfortable using these techniques.
You should not need any other information except the following (besides algebraic manipulations, and
integrals and derivatives of powers of x):
A. Maclaurin series for cos, sin, exponential (Note Maclaurin series is Taylor series at x = 0.)
cos(x)=1-.
sin(x) = x-
2! 4!
x³ x³
+ -
3! 5!
x³
e =1+x+3 +
2! 3!
7!
(h)n+¹ f(n+1) (c)
and c is some point
(n+1)!
B. Remainder formula for Maclaurin series: R₁(h) =
between 0 and h. Here R₁ (h) is the estimated error when the Maclaurin series is used up to the
n'th power of h.
Transcribed Image Text:Given that y(x) is a solution to yx² - 2y + 1 and y(0) = 1,
a. Find the first four terms in the Maclaurin series for y(x). (Hint: Write y(x) = ao +...+ a3x³ and
plug this expression into the differential equation. Then you may equate coefficients of x on both
sides of the equation.)
b. Estimate the error when the polynomial you found in (a) is used to estimate y(0.1). You may use
c = 0 in the remainder formula.
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