Appendix: All Solutions to the Problems
761
Solution
(1) The first variation of the functional is
δ J =
x 1
x 0
(aδy + bδy
+ 2cy
δy
)dx =
x 1
x 0
(aδydx + bdδy + 2cy
dδy
)
= (bδy + 2cy
δy
)
x 1
x 0
+
x 1
x 0
(aδydx − 2cy
δy
dx)
= ( bδy + 2cy
δy
− 2cy
δy)
x 1
x 0
+
x 1
x 0
[a + 2cy
(4)
]δydx
(2) The first variation of the functional is
δ J =
x 1
x 0
2y
1 + y 2 δy +
y 2 y δy
1 + y 2
dx
=
y 2 y δy
1 + y 2
x 1
x 0
+
x 1
x 0
2y
1 + y 2 −
(1 + y 2 )(2yy 2 + y 2 y ) − y 2 y 2 y
(1 + y 2 ) 3
δydx
=
y 2 y δy
1 + y 2
x 1
x 0
+
x 1
x 0
(1 + y 2 )(2y − y 2 y ) + y 2 y 2 y
(1 + y 2 ) 3
δydx
=
y 2 y δy
1 + y 2
x 1
x 0
+
x 1
x 0
y(2 − yy + 2y 2 )
(1 + y 2 ) 3
δydx
(3) The first variation of the functional is
δ J =
x 1
x 0
(2yδy − 2y
δy
− 2 cosh xδy)dx = 2
x 1
x 0
(yδy − y
δy
− cosh xδy)dx
= 2
x 1
x 0
(y + y
− cosh x)δydx − 2y
δy
x 1
x 0
(4) The first variation of the functional is
δ J =
x 1
x 0
(ax
2
δy
+ 2bx y
δy
)dx =
x 1
x 0
(ax
2
+ 2bx y
)dδy
= (ax
2
+ 2bx y
)δy
x 1
x 0
− 2
x 1
x 0
(ax + by
+ bx y
)δydx
(5) The first variation of the functional is
δ J =
x 1
x 0
(xδy + 2yδy − 4yy
δy − 2y
2
δy
)dx