Question

x < n with BCs y(0)= 0 and y(z) 0. (1 point) Find the eigenvalues and eigenfunctions for y" = Ay on 0 Note that any constant times an eigenfunction is also an eigenfunction. In order to obtain a unique solution find (x) so that x) dx 1 First find the eigenvalues and orthonormal eigenfunctions for n 1, i.e., An, >,(x). For n 0 there may or may not be an eigenpair. Give all these as a comma separated list. For example on an interval of length 1 if the eigenvalues are A with orthonormal eigenfunctions ,(x)= V2 sin(nax) then you would enter -(n pi)A2 sqrt(2 ) sin(n pi x) If in addition Ao = 0 is an eigenvalue with eigenfunction po(x) = 1 then you would enter 0, 1 -(n pi)^2 , sqrt(2 ) sin(n pi x)

Answer 1

consider y"-dy=0 for too PERt such that dapr. (OfJy zo mi przo a matr. nr Given that y(0)=0 =) Cita=of. * (T) 20 ) Cette Ce the =0 -Po career Toa) 670 then (=0. 3м v.344 , no eiзм чnctions ч Re Co% и for hao them an d a I= cxtla Given that yo] =0 (=0 1=0,2 Given that y(i)=0 = CiT=0 = 0,=0. : No soln. In the case also no eisen volnes s ain Cucine

Fordko then I pERt such that a= apm. Aegis niton=0 1 fix. y = C, cospate sinpx Given that yo) 20 = 9=0 in = sinpa Given that X(T)=0 = q sinpao final=0 th=nT for n=1,2,3. . pan. Gol 121,43, -a . The eisen values are A7 – 6 6 -1, +3, - - - the sespective eigen functions are ulxJ= sinne for n=1,2,3. -- Noco Tu pairs (-,sinx) (-4, sin2x), (-9, sinis), ...na (no, sinna?