Question

6. Determine the sequence of eigenvalues and corresponding eigenfunctions for each of the following Sturm-Liouville systems: (a) + Ap = 0, 9 (0) - 0) , 3º (2) + 5 °' (2) - 0) (b) 1 (0) + R -0, R'(1) - 0, R' (eº) - 0

Answer 1

ANSWER:

where y is a function of the free variable x. Here the functions p(x), q(x), and w(x) > 0 are specified at the outset. In the simplest of cases all coefficients are continuous on the finite closed interval [a,b], and p has continuous derivative. In this case, this function y is called a solution if it is continuously differentiable on (a,b) and satisfies the equation (1) at every point in (a,b). In addition, the unknown function y is typically required to satisfy some boundary conditions at a and b. The function w(x), which is sometimes also called r(x), is called the "weight" or "density" function.

6) a) t =0 6(a) a licos (Son) & Ensin (500) llojzo => cro 3442) + 54'(): 0 3 (2 sin (250) + 562 5T cos (250)=0 3 sin (25%) + SJL cos (15) tan (2.5x) +52 50 20 Joe eigen vawe s is the son of tanzł + 515020 S eigen fonction singa) lo aldr Lvda 1 +262 Rao rdidos colores +AR20 Rlar = ci cos(52 log 5) +(2 sin(50 loss) RIC)=0> 6250)

RleB). a) ci sin (503) ao 5x = 0 Asmens eigen valoe 21 or eigen vector = cos n% 10915)