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x²y" — x(x+2)y' + (x + 2)y = 0.

x²y" — x(x+2)y' + (x + 2)y = 0.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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4) Consider the ODE x²y″ − x(x + 2)y′ + (x + 2)y = 0. a) Use Abel's theorem to find W(y1, y2)(x) for any two solutions y₁(x) and y2(x) of the ODE. b) Use your answer from (a) to determine whether or not there can exist a fundamental set of solutions y₁(x) and y2(x) which are linearly independent on the whole real line. c) Verify that y₁(x): = x is one solution of the ODE. Find a second solution y2(x). Do NOT compute any series solutions for this question. d) For each point x0 = R (that is for all real xo) determine whether xo is an ordinary or singular point for the ODE. If it is a singular point, determine whether it is a regular or irregular singular point. Do NOT compute any series solutions for this question either.
Transcribed Image Text:4) Consider the ODE x²y″ − x(x + 2)y′ + (x + 2)y = 0. a) Use Abel's theorem to find W(y1, y2)(x) for any two solutions y₁(x) and y2(x) of the ODE. b) Use your answer from (a) to determine whether or not there can exist a fundamental set of solutions y₁(x) and y2(x) which are linearly independent on the whole real line. c) Verify that y₁(x): = x is one solution of the ODE. Find a second solution y2(x). Do NOT compute any series solutions for this question. d) For each point x0 = R (that is for all real xo) determine whether xo is an ordinary or singular point for the ODE. If it is a singular point, determine whether it is a regular or irregular singular point. Do NOT compute any series solutions for this question either.
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