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Unit 11: Sturm–Liouville’s Boundary Value Problems

11.5 Summary Notes

 The Sturm-Liouville’s boundary value problems leads us to eigenvalues and eigenfunctions
of certain second order differential equations.

 It is seen that the eigenfunctions form a set of orthonormal set and as so form a complete
set.

 This helps us in expanding a certain function in terms of eigenfunctions on an interval
(a, b).

11.6 Keywords

Bessel’s differential equations, Legendre differential equations and many more equations can be
written in the Sturm-Liouville equation.
Depending upon certain boundary conditions the solutions known as eigenfunctions can be
found that form orthogonal set.
11.7 Review Questions

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

y + y = 0, with y(0) = y = 0
2
2. Find all the eigenvalues and eigenfunctions of the Sturm-Liouville problem
y + y = 0, with y (0) = 3, y (c) = 0

Answers: Self Assessment

4
1. (xy ) y xy
x
2 2
2. (e x y ) 2 e x y 0
x
x
3. (x e y ) e y 0
1 cos x sin x cos 2x sin 2x
4. , , , , , ......
2
11.8 Further Readings

Books K. Yosida, Lectures in Differential and Integral Equations
Sneddon L.N., Elements of Partial Differential Equations

King A.C, Billingham J. and S.R. Otto, Differential Equations

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