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Unit 9: Adjoint and Self-Adjoint Equations

9.5 Summary Notes

 In this unit we rearrange certain linear equations of the second order in a way in which the
differential operator is self-adjoint.

 Examples of self-adjoint equations are Legendre equation, Bessel’s equations, Hermite
equations and many more.

 Putting these equations into self-adjoint form enables us to study certain properties known
as eigenvalue and eigenfunction expansions and completeness etc.

9.6 Keywords

Eigenfunctions are a set of solutions of the self-adjoint equations that form an orthonormal set
of complete system.
The real symmetric matrix is self-adjoint or an Hermitian operator.

9.7 Review Question

1. Show that
, ,
(xy (x)) = xy(x)
is self-adjoint on the interval (0, 1), with x = 0 a singular end point and x = 1 a regular end
point with the condition y(1) = 0.

9.8 Further Readings

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

Pipes L.A. and Harrill L.R., Applied Mathematics for Engineers and Physicists
Yosida K., Lectures on Differential and Integral Equations.

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