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y ₁ - 14 xy = 0 ower Series representations of element 8 = 2 = 1=1+x+ n=0 cosx= Σ n=0 2! (-1)"x2n (2n)! + x² 2! 4!

y ₁ - 14 xy = 0 ower Series representations of element 8 = 2 = 1=1+x+ n=0 cosx= Σ n=0 2! (-1)"x2n (2n)! + x² 2! 4!

Advanced Engineering Mathematics
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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Find Power series solution of the differential equation given below. Determine the radius of convergence of the resulting series, and use the series given below identify the series in terms of familar elementary. functions. to y ₁ - 14 xy = 0 Power Series representations of elementary functions 00 - Σ n=0 COS X = n! cosh x= 80 n=0 ∞0 sinx= Σ n=0 = 1+x+ In (1+x)= (-1)^x2n (2n)! 00 (-1)^x²n+1 (2n + 1)! sinhx= ) n=0 2n x² (2n)! n=0 ∞ 2n+1 +² 2! 00 = 1+ (2n + 1)! (1+x) = 1 + x + + n 1- (-1)^+ 1 x 3! =X- 2! =X+ α(α-1) 2! x² x4 2! 4! + + ... x³ 3! x³ x5 + 3! 5! X -x² + 4! + X- n=0 ∞0 1 1-²x = Σ x² = 1+x+x² + x³ + ... n=0 2 2 + ... 3 α(α-1)(x-2) 3! A) The power series solution is -x³+. y(x) = + (Please enter an expression in terms of C that includes all terms up to order 6.)
Transcribed Image Text:Find Power series solution of the differential equation given below. Determine the radius of convergence of the resulting series, and use the series given below identify the series in terms of familar elementary. functions. to y ₁ - 14 xy = 0 Power Series representations of elementary functions 00 - Σ n=0 COS X = n! cosh x= 80 n=0 ∞0 sinx= Σ n=0 = 1+x+ In (1+x)= (-1)^x2n (2n)! 00 (-1)^x²n+1 (2n + 1)! sinhx= ) n=0 2n x² (2n)! n=0 ∞ 2n+1 +² 2! 00 = 1+ (2n + 1)! (1+x) = 1 + x + + n 1- (-1)^+ 1 x 3! =X- 2! =X+ α(α-1) 2! x² x4 2! 4! + + ... x³ 3! x³ x5 + 3! 5! X -x² + 4! + X- n=0 ∞0 1 1-²x = Σ x² = 1+x+x² + x³ + ... n=0 2 2 + ... 3 α(α-1)(x-2) 3! A) The power series solution is -x³+. y(x) = + (Please enter an expression in terms of C that includes all terms up to order 6.)
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