8.8 Introduction to the Famous Scientists
509
(l + x)
∂
2 u
∂t 2 = a
2 ∂
2 u
∂ x 2
where, l is the length of the string, find the vibration frequency of the string
when both ends are fixed.
8.19 Find the characteristic value and characteristic function of the functional
J [y] =
e
1 (6y
2
+ x
2 y
2
)dx, such that it conforms to the boundary conditions
y(1) = y(e) = 0 and the isoperimetric condition
e
1 y
2 dx = 1.
8.20 Find the characteristic value and characteristic function of the functional
J [y] =
2π
π (y
2
− y
2
)dx, such that it conforms to the boundary conditions
y(π) = y(2π) = 0 and the isoperimetric condition
2π
π y
2 dx = 1.
8.21 Find the characteristic value and characteristic function of the functional
J [y] =
1
0 [3y
2
− (x + 1)
2 y
2
]dx, such that it conforms to the boundary
conditions y(0) = y(1) = 0 and the isoperimetric condition
1
0 y
2 dx = 1.
8.22 Find the characteristic value and characteristic function of the functional
J [y] =
3
0 [(2x + 3)y
2
− y
2
]dx, such that it conforms to the boundary
conditions y(0) = y(3) = 0 and the isoperimetric condition
3
0 y
2 dx = 1.