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2 Consider the IVP (x + 1)²y" – 2y = 0, y(0) = 1, y'(0) = -1. Determine all ordinary points of the differential equation. Verify that the solution of the IVP is given via $(x) = (-1)"x". %3D n=0 using the differentiation rule for power series and substituting the expression in the IVP. Determine the radius of convergence for its series solution y = ¢(x) about ro = 0. Using the geometric series show that y = (x + 1)¬1. Furthermore, determine a fundamental set of solutions of (x + 1)²y" – 2y = 0

2 Consider the IVP (x + 1)²y" – 2y = 0, y(0) = 1, y'(0) = -1. Determine all ordinary points of the differential equation. Verify that the solution of the IVP is given via $(x) = (-1)"x". %3D n=0 using the differentiation rule for power series and substituting the expression in the IVP. Determine the radius of convergence for its series solution y = ¢(x) about ro = 0. Using the geometric series show that y = (x + 1)¬1. Furthermore, determine a fundamental set of solutions of (x + 1)²y" – 2y = 0

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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Q.2 Consider the IVP (4) (x + 1)²y" – 2y = 0, y(0) = 1, y'(0) = -1. a) Determine all ordinary points of the differential equation. Verify that the solution of the IVP is given via (5) φ(τ)Σ- 1)"α". n=0 using the differentiation rule for power series and substituting the expression in the IVP. Determine the radius of convergence for its series solution y = 0(x) about xo = 0. b) Using the geometric series show that y = (x +1)-1. Furthermore, determine a fundamental set of solutions of (6) (x + 1)²y" – 2y = 0 via reduction of order. Verify that it is a fundamental set of solutions by computing its Wronskian W[y1, y2](x). Find the solution to the IVP (7) (x+ 1)²y" – 2y = 0, y(0) = 1, y'(0) = 1.
Transcribed Image Text:Q.2 Consider the IVP (4) (x + 1)²y" – 2y = 0, y(0) = 1, y'(0) = -1. a) Determine all ordinary points of the differential equation. Verify that the solution of the IVP is given via (5) φ(τ)Σ- 1)"α". n=0 using the differentiation rule for power series and substituting the expression in the IVP. Determine the radius of convergence for its series solution y = 0(x) about xo = 0. b) Using the geometric series show that y = (x +1)-1. Furthermore, determine a fundamental set of solutions of (6) (x + 1)²y" – 2y = 0 via reduction of order. Verify that it is a fundamental set of solutions by computing its Wronskian W[y1, y2](x). Find the solution to the IVP (7) (x+ 1)²y" – 2y = 0, y(0) = 1, y'(0) = 1.
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