Question

Show that the cigenvalue probom (ry' (r))' = Ary(r), 0 <r<R, y(0) is bounded, y(R) = 0 has no negative eigenvalues. Hint: Use an energy argument.

Answer 1

Auge Giren equ of radial is y"(+ fy'(o) = -14() 0 Multiplying by 8, Then -(ry'(o))' = Aoyco), ocr<R. — which is a Bessells equ of boundary condition. YCR) zo. The general solution of equo is y(x) = 4,50 (0) + C2%, (v), Here ro = vir - where, c, and cq are arbitoary constants and Jo() – esssel function of the second kind of order again. for this problem, we all given the bounadary condition as y(0)20 and ylo) is boundard: ie, lyco)<a or za number mo ly(o) .m.

We have the formula for Bessels fonction of the first and second kinds of order 'N' for x²yl taylt (21²_82)y zo. - Jo (1) 20 (-1) * ( vtak. kao k!l(k+v+0] 2) and Yv(x) = Jr (G) cos (v.fc) _J-1(2) sin (ra) we can observe that Jula) is fence as a o and Y, (x) is tending to o. e, Y, (0) 0 as ao. for any values of "V" here 1120, x=to=fir. we can say that Jo (ro) is finite as roo. and Yo (ro) -o as r o we have according to be boundary conditions y(o) is boundary condition. so y/o)=9,5. (Vo) + CqY (ro) where %=(150) The value of C has to be equal to zero ie C₂20 given Yolro) as roo.

which is the only possible way for the given Condition to be satisfied. Then from ie (9 (1) = c. Jo ro) +Cz Yo (ro). where yo (~). into y(x) = C, jolro), where ro-rout => Y) = ,5,68 Y), Also we are given y(1) 2o. here cijo (F1.8) 30 (Solr)) a to only when rr R by a not of Jo(SMR) evra jer ToCART CARJA : Jo (FIR) SI But we are told to consider the boundary. at ro tor=r. ie, 8" range over [0, R] ier takes only non-negative values.

so, that the not Yrir (for 5 (r) less info ne.r=8TR, &, is non-negative root to Joly) number »th= (he) Pin is non- negative root for 50 (0) observe that all xn's are squars of non- negative in's was always be non-negative There's no possibely for Xo's to be negative . since, the given eigen value problem ,6 has no negative eigen values.