Section 6.4Fourth Order Sturm-Liouville Theory101Exercises 6.41.This is a special case of Example 1 withL=2andλ=α4. The values ofαare the positive roots of the equationcos 2α=1cosh 2α.There are inFnitely many roots,αn(n=1,2,...), that can be approximated withthe help of a computer (see ±igure 1). To eachαncorresponds one eigenfunctionXn(x) = coshαnx-cosαnx-cosh 2αn-cos 2αnsinh 2αn-sin 2αn(sinhαnx-sinαnx).5.There are inFnitely many eigenvaluesλ=α4, whereαis a positive root of theequationcosα=1coshα.As in Example 1, the roots of this equation,αn(n,2), can be approxi-mated with the help of a computer (see ±igure 1). To eachαncorresponds oneeigenfunctionXn(x) = coshαnx-cosαnx-coshαn-cosαnsinhαn-sinαn(sinhαnx-sinαnx).The eigenfunction expansion off(x)=x(1-x), 0<x<1, isf(x∞±n=1AnXn(x),whereAn==²10x(1-x)Xn(x)dx²10X2n(x)dx.After computing several of these coeﬃcients, it was observed that:³10X2n(x)dx= 1 for alln,2,...,A2n= 0 for alln,2,....The Frst three nonzero coeﬃcients are
This is the end of the preview. Sign up
access the rest of the document.
This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.