Question

Find the eigenvalues in and eigenfunctions yn(x) for the given boundary-value problem. (Give your answers in terms of n, making sure that each value of n corresponds to a unique eigenvalue.) y" + y = 0, y(0) = 0, y(t) = 0 in = 1, 2, 3, ... în=0 Yn(x) = cos(nx) , n = 1, 2, 3, ... Need Help? Read It Talk to a Tutor

Answer 1

A 1 7 A - "+dy-0 9 (6) = 0, y (D = 0 Definition. Those values of t that give 5 non- trivial solutions, we call to an eigenvalue for the Boundary value problem. and the non-trivial solutions will be called eigenfunctions for the BVP, corresponding to the giren eigenvalue. we have three cases, 170, d=0, a co for each case, we will check, whether 3 the BVP for these to gires nontrivial solution E or not - so, letus start, D 120 then we have characteristic polynomial as - 82+2=0 = 8,2= 1 - 7 E srce, 170, these roots are complex. - 8,2 = 2 i then general solution. yw Glas (Jin) + C sin (1x) = on Applying Boundary condition, we get Or you = G & Geo and o- y (IT) = G (cas (7 7) + (2 sinan) → Cz sin(va T) co

> either go or sir tudo) 20. of gão, we will get trivial solution Yay To so (280 therefore, su (TTT) 50 - To=nT n= 1, 2, 3, .... - In =) de n2 , n = 1,2,3, en then, the positive ergervalues are then & on= n2, n= 4, 2, 3,111 and In (21) = sincurs, n = 1,2,3, 14, are eigen functions conesponding, dn, n = 1,2,3,4,6 es perral solution is 4 (01 to y(IT) = = Gotlar, Applying Boendary value & condition, we get y aro - G Y M = 0 = Cati u vão therefore d o gires trivial solution Hence deo is not eigenvalue.

In this case, as belous: characteristic equation is 22tico As sco, N., 2. are real, distinct roots, therefore, we have general solution as 2 fallours: y = Gcash (JTM) + la sish (Fox) s Applying Boundary condition, we get Gooi = 0 = 6 cash cost es ser hos = Golto 3 450 and & (T) = 0 = 4 cash (FIT) the sinh (1) o seich (PIT) , sine, gão - # either Geo Or sinh (J-f t to - But as I co, TJ-j to and so we have also serh Co-j ti to therefore, me have only one case left fil. Cão But this give trivial solution.

no Hence, for this BVP if we have do me get the trivial solition and so there are no negative eigenvalues, In summary, then, we will have the fallomig eigenvalues / eiger functions for this BUP. tu= 1 = n2, Inx) = sinnu), n = 1, 2, 3, m. N2