Discussion
Let \(p_{n}(x)\) be the polynomial solution of the differential equation
\[ \frac{d}{dx}[(1 - x^{2})y'] + n(n+1)y = 0 \]
with \(p_{n}(1) = 1\) for \(n = 1, 2, 3, \ldots\). If
\[ \frac{d}{dx}[p_{n+2}(x) - p_{n}(x)] = \alpha_{n} p_{n+1}(x), \]
then \(\alpha_{n}\) is
Answer: D
\(\alpha_{n} = 2n + 3\).
The equation is Legendre's equation, and its polynomial solution with \(p_{n}(1) = 1\) is the Legendre polynomial \(P_{n}\). The Legendre polynomials satisfy the recurrence
\[ P'_{m+1}(x) - P'_{m-1}(x) = (2m + 1)P_{m}(x) \]
Take \(m = n + 1\):
\[ P'_{n+2}(x) - P'_{n}(x) = (2n + 3)P_{n+1}(x) \]
Check with \(n = 1\), where \(P_{3} = \frac{1}{2}(5x^{3} - 3x)\):
\[ \begin{aligned} P'_{3} - P'_{1} &= \frac{15}{2}x^{2} - \frac{3}{2} - 1 \\ &= \frac{5}{2}(3x^{2} - 1) = 5P_{2} \end{aligned} \]
and \(2(1) + 3 = 5\). The values \(2n + 1\), \(2n\) and \(2n + 2\) give 3, 2 and 4 here, so they fail.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development