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y' cos(4x) + y, y(0) = 7 The Taylor solution is: y = ao + a1x + aɔx² + a3x³ + o(x*) where: ao =| 7 a1 = 50 a2 = 7 14 az =

y' cos(4x) + y, y(0) = 7 The Taylor solution is: y = ao + a1x + aɔx² + a3x³ + o(x*) where: ao =| 7 a1 = 50 a2 = 7 14 az =

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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## Differential Equation and Taylor Series Solution We are given the differential equation: \[ y' = \cos(4x) + y^2, \quad y(0) = 7 \] The Taylor series solution is expressed as: \[ y = a_0 + a_1x + a_2x^2 + a_3x^3 + o(x^3) \] ### Coefficients The values of the coefficients are: - \( a_0 = 7 \) (correct) - \( a_1 = 50 \) (correct) - \( a_2 = 7 \) (incorrect) - \( a_3 = -\frac{14}{6} \) ### Hint for Calculation Recall that the cosine function can be expanded using a Taylor series: \[ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \] This hint suggests how the cosine function in the given differential equation could be approximated in the Taylor series expansion.
Transcribed Image Text:## Differential Equation and Taylor Series Solution We are given the differential equation: \[ y' = \cos(4x) + y^2, \quad y(0) = 7 \] The Taylor series solution is expressed as: \[ y = a_0 + a_1x + a_2x^2 + a_3x^3 + o(x^3) \] ### Coefficients The values of the coefficients are: - \( a_0 = 7 \) (correct) - \( a_1 = 50 \) (correct) - \( a_2 = 7 \) (incorrect) - \( a_3 = -\frac{14}{6} \) ### Hint for Calculation Recall that the cosine function can be expanded using a Taylor series: \[ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \] This hint suggests how the cosine function in the given differential equation could be approximated in the Taylor series expansion.
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