Question

Question 4 5 pts Consider the equation nº X" () + X(T) - 0 with the following boundary conditions. X(0) + X'(0) -0, and X(n) + X'(n) - 0. 1. Write down the general solution to the equation for XC). 2. Write down the result of applying the boundary condition X(0) + X'(0) = 0 to the general solution 3. Write down the result of applying the remaining boundary condition X(n) - X'(n) = ( to the general solution. 4. Write down the resulting eigenvalues and the resulting eigenfunctions ()

n=5

Answer 1

Case I so, Given that ñ x²(x) + Xx (2) = X"(k) + W x (x) =0, when a=0, Then, X" (1) = the general salh X (W) ciute and X'(W) = a X(0) + X'(o) ate and X (1) + X(n)=0 tezo, - antata=0 (n+1)+ G=0. since mai to. So nti 1 9=₂=0 is the only sal". Therefore x=0, is not an eigen value case 2 When no aco, let À =- Ñ where K es positive หY X'(X) Ň X (W) the general Saln X(). un So ux M X'(U) ka Ge n now, using borendary condition. X(0) + X(0) = 0 cital + ₂-9 h = 0 att 4) + q (1-4) X (1) + X'(n) =0, alek + M M ) + Glék u (144) + 66k (1-4/2) = since 1- -un 1+ (1 h) - & ( - ) = (1-K) (6") a=₂=0 is the only selu Therefore x<0 is not an eigen value. , وه

case 3 then when ao X" (W) + 7 X (K) =0. the goneral Sain X (2) Gus an tosin on X'(x) = 1 sain 2x + 9 7 6 24 now using the boundary condition X(0) + X(0) = 0 X (1) + X (1) carrera 4+ 2 h = 0 aosat a sinx +(- Aqsina te 7 42) to 9=-6 la (essa - 2 sin 2) + 2 (sin x + 2 CD)=0 now, putting the value c in ean (2) we get - 22. 7 2 (cosa aa sinn) + G (sinat a cost) =0. - sinxt sinx + a nosa 7 n a sina (It since, a to lomer nuse (1=6=0 by 11) to, sina=0= sinna a=na whered n=1, 2, 3, nv The resulting elgonvalier an = na , 121,2,3,.. Putting the value a in 1) we get 9=-C2 NA GT n The resulting eigen fumetion Xx) Xn (0) (sinan -a s TX) - A cos nan 7 G sin nak