2.10 Invariance of the Euler Equation
185
when x 1 = x = 1, u 1 = 1. Therefore the original functional can be written as
J [y] =
u 1
u 0
v
u
v
v
2
− u
du
2
√
u
=
1
2
u 1
u 0
√
u(v
2
− v)du
(1)
The Euler equation of the functional is
−
√
u −
v
√
u
− 2
√
uv
= 0
( 2 )
or
v
+
1
2u
v = −
1
2
(3)
The integral factor of the Eq. (3) is e
1
2
du
u , thus its solution can be written as
v
√
u = c 1 −
1
2
√
udu
(4)
or
v
=
c 1
√
u
−
u
3
Integrate the above equation and back to the original variable, we obtain
y
2
= 2c 1 x −
x
4
6
+ c 2
(5)
The constants in accordance with boundary conditions are c 2 = 0, c 1 =
7
12
. So
the extremal curve is
y
2
=
7x − x
4
6
(6)
If staring directly from the given functional, then there is
2y(y
2
− x
2
) − 4yy
2
− 2y
2 y
= 0
( 7 )
or
yy
+ y
2
+ x
2
= 0
( 8 )
The above equation is recast as