Question

Consider the following boundary value problem, x2y′′ + 17xy′ + (64 + λ) y  =  0,    y(1)  =  0,  y(e6 )  =  0

(a)	Find the eigenvalues.

(b)

Find the eigenfunctions. Take the arbitrary constant (either c1 or c2) from the general solution to be 1. Consider the following boundary value problem, xy" + 17xy' + (64 + 2) y = 0, y(1) = 0, yle) = 0 (a) Find the eigenvalues. (b) Find the eigenfunctions. Take the arbitrary constant (either cy or c2) from the general solution to be 1.

Answer 1

Given is, i_ boundary value problem xy" + 17 xy' + (64+1)y 전 ylesize , y(i)=0 so we shall This is Pheed to find the roots of the Ender differential equation quadratic, seu n(n-1) + 18 + (64 + 2) = r²+16r + (64.+) = 0 4;(64 +1) ar cititles 2 d. 356 2 -16 +12VAN T 2 –8+ - Hi, Case 1: X>0 aire roots will be complex. so the roots Then = -8 t r The solution is -8 ci cos (en (a) ) + C2 y (a) sin (ln(x)) where c-G2 are constants. Now, y(i)=0.3.7. co.20 and y (eo) ao c e 48 cos (6VX) + c₂ e 4 sin (6ux)=0 & c. e 48 sin (6r2.) ao To avoid trivial solution sin (60) zo. 6a и пL t n=1,2, 36 nal,2,-- n²77² n. The eigen value is xna n² x² 36 the eigen functions Yn(x) = drile), n1,2,3 na 1, 2, and aure x sin MTL 6

case 2: d=0 Som Then the double root is y(x) = C,x-8 In Gae) ~=-8 and the solution is, ç, x how, 9 (1) 60 Ć,-0. and yle") so 60e48 6 eal at (2 co ܚܐ since we have only trivial solution it for this eigen value case not an aco SO Case 3 so 시 Then two real and distinct roots are r=-8&F and the solution is -84. 8 -- Y(x) = C, X t.cz re Now, Then yll) 20 ☆ C, + C2 20: * Cuisitic, the solution becomes, -872 y ( ) = -५४ +6 yles),50 ce C, X c c; x 8-a -486FX Now cie -48 e. leon -6 е. ). 4 since the exponents on the -त्र #C, two terms in the are not the same so C,0 =62 only the trivial solution and valines for izo. paran thesis and have so we there are no eigen