318
4 Problems with Variable Boundaries
4.20 For the extremal problem of the functional J [y] =
x 1
0 (y
4
− 6y
2
)dx, is there
any solution with the corner point? The boundary conditions are y(0) = 0,
y(x 1 ) = y 1 .
4.21 Find the transversality condition of the functional J [y]
=
x 1
x 0
f (x, y)e
arctan y
1 + y 2 dx.
4.22 Using the necessary condition δ J = 0 of a functional obtaining extremum,
find the function that can make the functional J [y] =
1
0 (y
2
− 2x y)dx reach
extremum. The boundary conditions are y(0) = y
(0) = 0, y(1) =
1
120
, y
(1)
is not given.
4.23 Under the condition that the admissible curve can not pass through the interior
of the circle domaim surrounded by the circumference (x − 5)
2
+ y
2
= 9, find
the curve that can make the functional J [y] =
10
0 y
3 dx reach exextremum,
the boundary conditions are y(0) = y(10) = 0.
4.24 Find the function that can make the functional J [y] =
π
4
0 (y
2
− y
2
)dx reach
extremum, one boundary point is fixed, y(0) = 0, another boundary point can
slide on the straight line x =
π
4
.
4.25 Only using the necessary condition δ J = 0, find the curve that can make the
functional J [y] =
x 1
0
√
1+y 2
y
dx attain extremum, one boundary point is fixed,
y(0) = 0, another boundary point (x 1 , y 1 ) can move on the circumference
(x − 9)
2
+ y
2
= 9.
4.26 A moving point lands from point A(x 0 , y 0 ) outside the curve y = ϕ(x) to
point B(x 1 , y 1 ) on the curve, what should the shortest time to take be? Known
when the moving point moves outside of the curve y = ϕ(x), the velocity is a
constant and equal to v 1 ; But it moves along the curve y = ϕ(x), the velocity
is also a constant and equal to v 2 , and v 2 > v 1 .
4.27 Find the extremal curve of the functional J =
x 1
0
y
−y
2 tan ϕ
y +tan ϕ
(ax + b)dx, where
one endpoint is undetermined on the y axis, another point is fixed, y(x 1 ) = 0.
4 Problems with Variable Boundaries
4.20 For the extremal problem of the functional J [y] =
x 1
0 (y
4
− 6y
2
)dx, is there
any solution with the corner point? The boundary conditions are y(0) = 0,
y(x 1 ) = y 1 .
4.21 Find the transversality condition of the functional J [y]
=
x 1
x 0
f (x, y)e
arctan y
1 + y 2 dx.
4.22 Using the necessary condition δ J = 0 of a functional obtaining extremum,
find the function that can make the functional J [y] =
1
0 (y
2
− 2x y)dx reach
extremum. The boundary conditions are y(0) = y
(0) = 0, y(1) =
1
120
, y
(1)
is not given.
4.23 Under the condition that the admissible curve can not pass through the interior
of the circle domaim surrounded by the circumference (x − 5)
2
+ y
2
= 9, find
the curve that can make the functional J [y] =
10
0 y
3 dx reach exextremum,
the boundary conditions are y(0) = y(10) = 0.
4.24 Find the function that can make the functional J [y] =
π
4
0 (y
2
− y
2
)dx reach
extremum, one boundary point is fixed, y(0) = 0, another boundary point can
slide on the straight line x =
π
4
.
4.25 Only using the necessary condition δ J = 0, find the curve that can make the
functional J [y] =
x 1
0
√
1+y 2
y
dx attain extremum, one boundary point is fixed,
y(0) = 0, another boundary point (x 1 , y 1 ) can move on the circumference
(x − 9)
2
+ y
2
= 9.
4.26 A moving point lands from point A(x 0 , y 0 ) outside the curve y = ϕ(x) to
point B(x 1 , y 1 ) on the curve, what should the shortest time to take be? Known
when the moving point moves outside of the curve y = ϕ(x), the velocity is a
constant and equal to v 1 ; But it moves along the curve y = ϕ(x), the velocity
is also a constant and equal to v 2 , and v 2 > v 1 .
4.27 Find the extremal curve of the functional J =
x 1
0
y
−y
2 tan ϕ
y +tan ϕ
(ax + b)dx, where
one endpoint is undetermined on the y axis, another point is fixed, y(x 1 ) = 0.