2.11 Introduction to the Famous Scientists
195
2.18 Find the extremal curve of the functional J [y] =
x 1
x 0
y−xy
y 2 dx, the boundary
conditions are y(x 0 ) = a, y(x 1 ) = b.
2.19 Discuss whether the variational problem of the functional J [y] =
π
2
0
x sin y +
1
2
x
2 y
cos y
dx makes sense, the boundary conditions are
y(0) = 0, y
π
2
=
π
2
.
2.20 Discuss the variational problem of the functional J [y]
=
π
2
0 (y
2
+ y
sin 2x)dx, the boundary conditions are y(0) = k, y
π
2
= −1.
What value does k take its extremal curve exists? and calculate the extremal
curve.
2.21 Discuss the extremum of the functional J [y] =
x 1
x 0
[2x y + (x
2
+ e
y
)y
]dx,
the boundary conditions are y(x 0 ) = y 0 , y(x 1 ) = y 1 .
2.22 Discuss the extremum of the functional J [y] =
1
0 (e
y
+ x y
)dx, the boundary
conditions are y(0) = 0, y(1) = a.
2.23 Discuss the extremum of the functional J [y] =
x 1
x 0
(y
2
+ 2x yy
)dx, the
boundary conditions are y(x 0 ) = y 0 , y(x 1 ) = y 1 .
2.24 Discuss the extremum of the functional J [y] =
1
0 (x y + y
2
− 2y
2 y
)dx, the
boundary conditions are y(0) = 1, y(1) = 2.
2.25 Find the extremal curve of the functional J [y] =
x 1
x 0
y
(1 + x
2 y
)dx.
2.26 Find the extremal curve of the functional J [y] =
2
1 x
2 y
2 dx, the boundary
conditions are y(1) = 1, y(2) =
1
2
.
2.27 Find the extremal curve of the functional J [y] =
1
0 (x + y
2
)dx, the boundary
conditions are y(0) = 1, y(1) = 2.
2.28 Find the extremal curve of the functional J [y] =
x 1
x 0
y
(1 + x
2 y
)dx.
2.29 Find the extremal curve of the functional J [y] =
x 1
x 0
(x y
+ y
2
)dx.
2.30 Find the extremal curve of the functional J [y] =
x 1
x 0
x
n y
2 dx, the boundary
conditions are y(x 0 ) = y 0 , y(x 1 ) = y 1 .
2.31 Find the extremal curve of the functional J [y] =
x 1
x 0
y
2
x k dx, where, k > 0.
2.32 Find the extremal curve of the functional J [y] =
π
0 y
2 sin xdx.
2.33 Find the extremal curve of the functional J [y] =
x 1
x 0
x
x+y dx.
2.34 Find the extremal curve of the functional J [y] =
x 1
x 0
√
1+y 2
x+k
dx, where, k is a
constant.
2.35 Find the extremal curve of the functional J [y] =
x 1
x 0
(ax + b)
1 + y 2 dx.
2.36 Find the extremal curve of the functional J [y] =
x 1
x 0
e
x
1 + y 2 dx.
2.37 If the first quadrant is filled with a transparent optical medium, and in any point
of which the velocity of light is equal to 1 + x, find the light path equation that
the light beam spreads from the origin to point (2, 3) in the shortest time.
2.38 Let F ≡ F(y
), prove that the extremal curve of the functional J [y] =
x 1
x 0
F(y
)dx is a straight line.
2.39 Find the extremal curve of the functional J [y] =
2
1 (1 + y
)
2
(1 − y
)
2 dx, the
boundary conditions are y(1) = 1, y(2) =
1
2
.
2.40 Find the extremal curve of the functional J [y] =
x 1
x 0
1 + y 2 y 2 dx.
195
2.18 Find the extremal curve of the functional J [y] =
x 1
x 0
y−xy
y 2 dx, the boundary
conditions are y(x 0 ) = a, y(x 1 ) = b.
2.19 Discuss whether the variational problem of the functional J [y] =
π
2
0
x sin y +
1
2
x
2 y
cos y
dx makes sense, the boundary conditions are
y(0) = 0, y
π
2
=
π
2
.
2.20 Discuss the variational problem of the functional J [y]
=
π
2
0 (y
2
+ y
sin 2x)dx, the boundary conditions are y(0) = k, y
π
2
= −1.
What value does k take its extremal curve exists? and calculate the extremal
curve.
2.21 Discuss the extremum of the functional J [y] =
x 1
x 0
[2x y + (x
2
+ e
y
)y
]dx,
the boundary conditions are y(x 0 ) = y 0 , y(x 1 ) = y 1 .
2.22 Discuss the extremum of the functional J [y] =
1
0 (e
y
+ x y
)dx, the boundary
conditions are y(0) = 0, y(1) = a.
2.23 Discuss the extremum of the functional J [y] =
x 1
x 0
(y
2
+ 2x yy
)dx, the
boundary conditions are y(x 0 ) = y 0 , y(x 1 ) = y 1 .
2.24 Discuss the extremum of the functional J [y] =
1
0 (x y + y
2
− 2y
2 y
)dx, the
boundary conditions are y(0) = 1, y(1) = 2.
2.25 Find the extremal curve of the functional J [y] =
x 1
x 0
y
(1 + x
2 y
)dx.
2.26 Find the extremal curve of the functional J [y] =
2
1 x
2 y
2 dx, the boundary
conditions are y(1) = 1, y(2) =
1
2
.
2.27 Find the extremal curve of the functional J [y] =
1
0 (x + y
2
)dx, the boundary
conditions are y(0) = 1, y(1) = 2.
2.28 Find the extremal curve of the functional J [y] =
x 1
x 0
y
(1 + x
2 y
)dx.
2.29 Find the extremal curve of the functional J [y] =
x 1
x 0
(x y
+ y
2
)dx.
2.30 Find the extremal curve of the functional J [y] =
x 1
x 0
x
n y
2 dx, the boundary
conditions are y(x 0 ) = y 0 , y(x 1 ) = y 1 .
2.31 Find the extremal curve of the functional J [y] =
x 1
x 0
y
2
x k dx, where, k > 0.
2.32 Find the extremal curve of the functional J [y] =
π
0 y
2 sin xdx.
2.33 Find the extremal curve of the functional J [y] =
x 1
x 0
x
x+y dx.
2.34 Find the extremal curve of the functional J [y] =
x 1
x 0
√
1+y 2
x+k
dx, where, k is a
constant.
2.35 Find the extremal curve of the functional J [y] =
x 1
x 0
(ax + b)
1 + y 2 dx.
2.36 Find the extremal curve of the functional J [y] =
x 1
x 0
e
x
1 + y 2 dx.
2.37 If the first quadrant is filled with a transparent optical medium, and in any point
of which the velocity of light is equal to 1 + x, find the light path equation that
the light beam spreads from the origin to point (2, 3) in the shortest time.
2.38 Let F ≡ F(y
), prove that the extremal curve of the functional J [y] =
x 1
x 0
F(y
)dx is a straight line.
2.39 Find the extremal curve of the functional J [y] =
2
1 (1 + y
)
2
(1 − y
)
2 dx, the
boundary conditions are y(1) = 1, y(2) =
1
2
.
2.40 Find the extremal curve of the functional J [y] =
x 1
x 0
1 + y 2 y 2 dx.