Find all real eigenvalue and eigenfunctions to the following: (xy')' + rx^(-1)y = 0 y'(0)=0 ; y(e^pi)=0 This is problem #19 to Fundamentals of Differential Equations and Boundary Value Problems 6th Edition.

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x2y"+xy'+ry=0 a2+r=0 r should be w2 so a=+-j(or i) y_1=coshlnwx y_2=sinhlnwx

y'(1)=0 so y_1 is eigen function y(epi)=0 so cospiw=0 then w=n-1/2 then r=(n-1/2)^2 are eigenvalue. That's it !

and fix your question y"(0)=0 change to y'(1)=0 !

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