Differential Equations and Their Applications pp 514-544 | Cite as

# Sturm-Liouville boundary value problems

Chapter

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## Abstract

In Section 5.5 we described the remarkable result that an arbitrary piecewise differentiable function
or a pure cosine series of the form
on the interval 0 <
and
Recall that Equations (3) and (4) have nontrivial solutions
respectively, only if

*f*(*x*) could be expanded in either a pure sine series of the form$$f\left( x \right) = \sum\limits_{n = 1}^\infty{{b_n}\sin \frac{{n\pi x}}{l}} $$

(1)

$$f\left( x \right) = \frac{{{a^0}}}{2}\sum\limits_{n = 1}^\infty{{a_n}\cos \frac{{n\pi x}}{l}} $$

(2)

*x*<*l*. We were led to the trigonometric functions appearing in the series (1) and (2) by considering the 2 point boundary value problems$$y'' + \lambda y = 0,y\left( 0 \right) = 0,y\left( l \right) = 0,$$

(3)

$$y'' + \lambda y = 0,y'\left( 0 \right) = 0,y'\left( l \right) = 0.$$

(4)

$${y_n}\left( x \right) = c\sin \frac{{n\pi x}}{l}and{\kern 1pt} {y_n}\left( x \right) = c\cos \frac{{n\pi x}}{l},$$

*λ*=*λ*_{n}=*n*_{2}*π*_{2}/*l*_{2}. These special values of*λ*were called eigenvalues, and the corresponding solutions were called eigenfunctions.## Keywords

Nontrivial Solution Product Space Orthogonal Basis Hermitian Operator Laguerre Polynomial
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1993