23 Strum Liouvill
Consider the DE with scalar \(\lambda\) defined in \([a,b]\)
\[ \frac{d}{dx} \Big[ P(x) y' \Big] + \Big[\lambda Q(x) + R(x) \Big] y = 0 \]
with the boundary conditions
\[ \begin{aligned} c_1 y(a) + c_2 y'(a) &= 0 & d_1 y(b) + d_2 y'(b) &= 0 \\ c_1 \text{ or } c_2 &= 0 & d_1 \text{ or } d_2 &= 0 \end{aligned} \]
Simplest Form¶
\[ \begin{aligned} y'' + \lambda y &= 0 \\ P(x) &= 1 \\ Q(x) &= 1 \\ R(x) &= 0 \end{aligned} \]
Legendre Equation¶
Legendre Equation can be represented as Strum-Liouvile Problem.
\[ \frac{d}{dx} \Big[ \underbrace{(1-x^2)}_{P(x)} y' \Big] + \underbrace{n(n+1)}_{\lambda} \ y = 0 \\ P(x) = 1-x^2 \\ Q(x) = 1 \\ R(x) = 0 \\ \lambda = n(n+1) \]
Here, \(\lambda\) is the eigen value of equation
The corresponding solutions are \(P_n(x), n = 1, 2, \dots\) They are called as eigen functions.
\(n > 0\) because \(n \le 0\) will give trivial solution.
Eigen Value/Function¶
\[ \begin{aligned} y'' + \lambda y &= 0 \\ y(a) &= 0 \\ y(b) &= 0 \\ a & \ne b \end{aligned} \]