Page 221 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Self Assessment

3. Find the eigenvalues and eigenfunctions of the equation
2
d y y = 0
dx 2
when y (0) = 0, y ( ) = 0
Show that the eigenfunctions are orthogonal to each other.

13.5 Summary

 In this unit we have review some of the properties of the solutions of equations like Bessel
equations, Legendre equations, Hermite equations and Laguerre equations which are of
Sturm-Liouville’s form.

 This way we can construct the eigenfunctions for certain eigenvalues of other equations
which resemble Sturm-Liouville problem with certain boundary conditions.

13.6 Keywords

Eigenfunctions are solutions of Sturm-Liouville problem corresponding to certain values of the
parameter called the eigenvalues.
Sturm-Liouville boundary value problem helps us to find eigenvalues and eigenfunctions in a
systematic way and their properties are well understood.

13.7 Review Questions

1. Find the eigenvalues and eigenfunctions of the Sturm-Liouville problem

y y 0, y (0) y 0
2
2. Show that the given set is orthogonal on the given interval and determine the corresponding
orthonormal set

1, cos x, cos 2x, cos 3x, ..., 0 x
Answers: Self Assessment

1. A = 0, B = 3
1 1
2. K n , y n ( ) A n sin n x , n 0,1,2,....
x
2 2
3. n 2 , y n ( ) sin nx n 1,2,3,....
,
x
13.8 Further Readings

Books Yosida, K., Lectures in Differential and Integral Equations
King A.C., Billingham, J. and Otto S.R., Differential Equations

214 LOVELY PROFESSIONAL UNIVERSITY