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3. Consider the 2nd order linear ODE: (1-x²)y" - xy' + a²y = 0 (1) where a is a real constant. (a) Solve this equation about the ordinary point xo = 0 by finding a power series solution, con- vergent for x < 1 by: (i) finding the recurrence relation and (ii) determining two linearly independent solutions y₁, y2 (write out the first four terms of y₁ and y2). (b) Now assume a = 2. Show that one of the linearly independent solutions is a polynomial (i.e. the series terminates). Do the same for a = 3. Using these results, argue that if a equals a positive integer n, then one of the solutions to (1) will be a polynomial of degree n.

3. Consider the 2nd order linear ODE: (1-x²)y" - xy' + a²y = 0 (1) where a is a real constant. (a) Solve this equation about the ordinary point xo = 0 by finding a power series solution, con- vergent for x < 1 by: (i) finding the recurrence relation and (ii) determining two linearly independent solutions y₁, y2 (write out the first four terms of y₁ and y2). (b) Now assume a = 2. Show that one of the linearly independent solutions is a polynomial (i.e. the series terminates). Do the same for a = 3. Using these results, argue that if a equals a positive integer n, then one of the solutions to (1) will be a polynomial of degree n.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 12CR
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3. Consider the 2nd order linear ODE: (1 — x²)y" - xy' + a²y = 0 (1) where a is a real constant. (a) Solve this equation about the ordinary point äî = 0 by finding a power series solution, con- vergent for x < 1 by: (i) finding the recurrence relation and (ii) determining two linearly independent solutions y₁, y2 (write out the first four terms of y₁ and y2). (b) Now assume a = 2. Show that one of the linearly independent solutions is a polynomial (i.e. the series terminates). Do the same for a = 3. Using these results, argue that if a equals a positive integer n, then one of the solutions to (1) will be a polynomial of degree n.
Transcribed Image Text:3. Consider the 2nd order linear ODE: (1 — x²)y" - xy' + a²y = 0 (1) where a is a real constant. (a) Solve this equation about the ordinary point äî = 0 by finding a power series solution, con- vergent for x < 1 by: (i) finding the recurrence relation and (ii) determining two linearly independent solutions y₁, y2 (write out the first four terms of y₁ and y2). (b) Now assume a = 2. Show that one of the linearly independent solutions is a polynomial (i.e. the series terminates). Do the same for a = 3. Using these results, argue that if a equals a positive integer n, then one of the solutions to (1) will be a polynomial of degree n.
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