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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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 50E
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Please do part A and B and please show steps and explain

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: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.
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.
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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