Question

Plz solve Part (B) & Part (C) with all the detailed clear steps and bcz I don't understand them at all

EXERCISES 5.5 5.5.1. A Sturm-Liouville eigenvalue problem is called self-adjoint if b dv dx du dac = 0 р u a because then SuL(v) - VL(u)] da = 0 for any two functions u and v satisfying the boundary conditions. Show that the following yield self-adjoint problems: (a) 7(0) = 0 and o(L) = 0 (b) (0) = 0 and $(L) = 0 (c) d (0) - hợ(0) = 0 and d (L) = 0 (d) o(a) = 4(b) and pla) (a) = p(b) () (e) º(a) = 6(b) and (a) = d.o. () (self-adjoint only if p(a) = p(b)] 4 No more than two independent eigenfunctions are possible, since the differential equation is of second order.

i need it in 4-8 hrs plz

with confident sol

Answer 1

The . do b) Objective is show that dll lolo and $(1) 0 Yield Sily adjoint problem let u and all are any that Satisfy the following boundary Conditions, to functions dolo) 20 (1) da let exaluate plu. du moodu djli da 2P (L) duty wir bers) - را به lucy (ulo, delo v to 4 and we beliefly the bowodany Condition (i] bo da dulo da Plo) (ulol: > (6) 20) 4 w(L): 0 Since dulo) 14(L) -0, dulos do Thus, Eqn (2) become p The above result shows that the boundary condition (1) Yeild Self- adjoint problem. -O da

Conditions. (4)=0 do dy vi Be Objective is show that do (o)- hello)= of finde ufy Satisfy the boundary Condition (i), Lo. do do (4=0 Nield do Self-adjoint problem. let u fy are i to functions that Satisfy the any following boundary do 10). bolo 30 161) let Evaluate aplo) dulo), bulo), duchy dulo da bubo), and du(e) Tar do Prus, Eqn (a) becomes. Pludy 0-PG9 [wtoyhero; thulay wel]>oplating The above result shows that the boundary conditions (7) Meild. Self-adjoint problem. da da da (2) da