Answer to Question #203759 in Differential Equations for Rama

Question

1. The eigenfunction equation X00 = -λX is regular since s(x)ρ(x) = 1 , 0 for all x.

2. Bessel’s equation is regular for x 2 [1; 2].

3. Bessel’s equation is singular for x 2 [0; 1].

1. Solve the Sturm-Liouville equation X00 = -λX on [0; π] with data X(0) = 0 and X(π) = 0,

Ans: eigenfunction X(x) = sin(nx) eigenvalue λ = n2, for any positive integer n.

2. Solve the Sturm-Liouville equation X00 = -λX on [0; L] with data X(0) = 0 and X0(L)+hX(L) =

0.

( Ans: eigenfunction X(x) = sinx pλ
eigenvalue λ given by a positive solution of the transcendental equation pλ + h tanL pλ
= 0.

Answer 1

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