Find three positive values of $\lambda$ for which the differential equation $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$ has polynomial solutions.
A) 2, 6, 12
B) 1, 2, 3
C) 1, 4, 9
D) 2, 4, 6
E) 2, 6, 10

Question

Quizplus · Accepted Answer

The differential equation given is a form of the Legendre's differential equation, which has polynomial solutions for $\lambda = n(n + 1)$ where $n$ is a non-negative integer. The first three positive values of $\lambda$ using this formula are for $n = 1, 2, 3$, which give $\lambda = 2, 6, 12$ respectively.

Find Three Positive Values Of λ\lambdaλ For Which the Differential Equation (1−x2)y′′−2xy′+λy=0\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0(1−x2)y′′−2xy′+λy=0

Find Three Positive Values Of $\lambda$ For Which the Differential Equation $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$