Consider the following boundary value problem, r?y" + 19xy' + (81 +2) y = 0, y(1)...

ouden Given boundary value problem, re?y" + 19 xy' + (81+d) y =0 ; 901) = 0, y (es) = This is the homogeneous dit gesiential equation of and Then, this {DCD-1) + + 101 +4)3 = 0 + z = ln (oc) Auxiliary Equation of this D.E can wuitten as ; be 19 D Z where na e mcm-1) +19 m + Oltd ma 2 mt 19 m + 01 to m+ 18 m+ 1 + Case-I Take d, m2 + lom + OLEO (m + 9)2 = 0 -9 (z) -ga се + Cg ze -ga (c + C2 z) e {cit ca en ca)} & - 9 -92 - 9 ayx) since you = 0 + CI y ces) & (CI + Cax C2 = 0 :6) (@sja y(x) =0 Hence, d=0 is not and eigenvalue Case - II all , Take d = a m? + 10m + 01 + a -10 of 2 (10)2 - 4 (01 792) m =

-10 + 25-02 -10 + 20 lo 2 ma -9 + a Then, - Iz ycz)= e { cos Caz) uls + C2 CI COS sin (az)} yox) { ci cos 9 xe sla ln (x)) + cp sin(a locx))} since, you) =0 C = 0 pub Ci cos o y (eo) Ceb) = 0 =0 Ces jo {cos((6)) + Ca sin(96))} و روم) sin coa) Ca sin coa) =0 {"' C203 ба n TT a = NTT 8 Then, sin y (x) Са nt enca) 6 { 6 {rezug u} uls og roponubia no X २ ㅋ nat SI (noun) Case- P 2 a an 36 Take 9 18 m 92 +01 -10 I J(10) (18)? -4601-92) -10+ 2a 2 -9 ta + ш 1 m - ma

Thin, (-9+a) z y(z) = (-g-a) Z + ca e -cata) CI X + C2 since year goo) C (1) 9-9 - (a +9) + Cg (1) and y (eo) - 6cato) CI 669-9) е + ca 9c2=0 Therefore, 0 y (x) an de-? is not eigenvalue Hence, (a) Eigenvalue of given boundary value problem is d . na? 36 no 0,1,2,... ane Eigenfunctions 1 g(x) sin {nh n T ln (x)} 9 x