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2. Laguerre polynomials arise in quantum mechanics and are the solutions of Laguerre's equation xy" + (1 – x)y' + ny = 0 for each n e Z (n > 0). The first four Laguerre polynomials (n = 0, 1, 2, 3) are 1, 1- x, 2 - 4x + x², 6 – 18r + 9x? – x³. Recall that P3 denotes the real vector space of all polynomials of degree at most 3. (a) Determine whether the first four Laguerre polynomials span P3. In your solution you must use the definition of span to derive the equations to be solved. (b) Do the first four Laguerre polynomials form a basis for P3? Explain your answer.

2. Laguerre polynomials arise in quantum mechanics and are the solutions of Laguerre's equation xy" + (1 – x)y' + ny = 0 for each n e Z (n > 0). The first four Laguerre polynomials (n = 0, 1, 2, 3) are 1, 1- x, 2 - 4x + x², 6 – 18r + 9x? – x³. Recall that P3 denotes the real vector space of all polynomials of degree at most 3. (a) Determine whether the first four Laguerre polynomials span P3. In your solution you must use the definition of span to derive the equations to be solved. (b) Do the first four Laguerre polynomials form a basis for P3? Explain your answer.

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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2. Laguerre polynomials arise in quantum mechanics and are the solutions of Laguerre's equation xy" + (1 – x)y' + ny = 0 for each n e Z (n > 0). The first four Laguerre polynomials (n = 0, 1, 2, 3) are 1, 1- x, 2 – 4x + x², 6 – 18.x + 9x² – x³. Recall that P3 denotes the real vector space of all polynomials of degree at most 3. (a) Determine whether the first four Laguerre polynomials span P3. In your solution you must use the definition of span to derive the equations to be solved. (b) Do the first four Laguerre polynomials form a basis for P3? Explain your answer.
Transcribed Image Text:2. Laguerre polynomials arise in quantum mechanics and are the solutions of Laguerre's equation xy" + (1 – x)y' + ny = 0 for each n e Z (n > 0). The first four Laguerre polynomials (n = 0, 1, 2, 3) are 1, 1- x, 2 – 4x + x², 6 – 18.x + 9x² – x³. Recall that P3 denotes the real vector space of all polynomials of degree at most 3. (a) Determine whether the first four Laguerre polynomials span P3. In your solution you must use the definition of span to derive the equations to be solved. (b) Do the first four Laguerre polynomials form a basis for P3? Explain your answer.
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