Find the eigenvalues lambdan and eigenfunctions lambdanx for

Find the eigenvalues lambda_n and eigenfunctions lambda_n(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\" + lambda y = 0, k(0) = 0, y(pi/5) = 0 lambda n =, n = 1, 2, 3,... y_n(x) =, n = 1, 2, 3,...

Solution

Given eqn -

y\'\' + y = 0; y(0) = 0 , y\'(/2) = 0

The general solution is -

y(x) = a cos( x ) + b sin ( x) ------- (1)

differentiating the 1st eqn,we get

y\'(x) = - a* sin ( x ) + b* cos ( x ) -----------(2)

then apply the first condition,

putting x=0 and y=0 in eqn 1

we have y(0) = 0

we get a = 0 ---------(3)

apply the second condition,

putting x=/2 and y=0

we have, y\'( /2 ) = 0 also a=0

we get,

b cos ( */2 ) = 0

now either b = 0 or cos ( /2* ) =0

but if b=0,

we get a trivial solution i.e.. y =0 since a & b both are zero.

hence,

cos ( * /2 ) = 0

hence, * /2 = * ( 2n + 1 ) / 2

  _n= (2n+1)²

Since we have a = 0, only sine term remains, so eigenfunctions are

y_n = bsin( _nx) for any b

with eigenvalues

_n = ( 2n + 1 )² for n = 1,2,3........