Question

If
\[ y=e^{a\sin^{-1}x}=(1-x^2)y_{n+2}-(2n+1)xy_{n+1} \] is equal to

• A. \(( -n^2+a^2 )y_n\)
• B. \(( n^2-a^2 )y_n\)
• C. \(( n^2+a^2 )y_n\)
• D. \(( -n^2-a^2 )y_n\)

Hint:

For \(y=e^{a\sin^{-1}x}\), recurrence relation is \((1-x^2)y_{n+2}-(2n+1)xy_{n+1}=(n^2+a^2)y_n\).

Answer 1

The Correct Option is C

Step 1: Recognize standard recurrence relation.
For functions of the form \(y=e^{a\sin^{-1}x}\), derivatives satisfy a known recurrence:
\[ (1-x^2)y_{n+2}-(2n+1)xy_{n+1}=(n^2+a^2)y_n \]
Step 2: Use this identity directly.
The given expression matches the LHS.
Hence RHS equals:
\[ (n^2+a^2)y_n \]
Final Answer:
\[ \boxed{(n^2+a^2)y_n} \]