8.7 Eigenvalues and Eigenfunctions of Operator Equations
495
The corresponding eigenfunction is
y 21 = (1 − x
2
)
343 + 28
√
133
−147
+ x
2
a 2
(10)
For the second eigenvalue of λ
2 , there is
a 1 =
7 − 2λ
2
2
14λ
2
2 − 35
a 2 =
343 − 28
√
133
−147
a 2
(11)
The corresponding eigenfunction is
y 22 = (1 − x
2
)
343 − 28
√
133
−147
+ x
2
a 2
(12)
For this example, the general solution of differential equation is y = c 1 cos λx +
c 2 sin λx, according to the boundary conditions, there are
c 1 cos λ − c 2 sin λ = 0
c 1 cos λ + c 2 sin λ = 0
(13)
If the homogeneous equations (13) have the nonzero solution, then the determinant
of coefficients should be zero, namely
cos λ − sin λ
cos λ sin λ
= 2 sin λ cos λ = sin 2λ = 0
( 1 4 )
2λ = nπ can be obtained, or λ n =
nπ
2
. Thus the first eigenvalue is λ
2
1 =
π
2
2 , the
second eigenvalue corresponding to the approximate solution is the third eigenvalue
λ
2
3 =
3π
2
2 of the analytic solution.
The relative error of the first eigenvalue is
E 1 =
14 −
√
133 −
π
2
2
π
2
2
× 100% = 0.001471%
(15)
The relative error of the second eigenvalue is
E 2 =
14 +
√
133 −
3π
2
2
3π
2
2
× 100% = 14.977309%
(16)