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In the study of ordinary differential equations and their associated boundary value problems , Lagrange's identity gives the boundary terms arising from integration by parts of a self-adjoint linear differential operator . Lagrange's identity is fundamental in Sturm–Liouville theory . In more than one independent variable, Lagrange's identity is generalized by Green's second identity .
Statement
Lagrange's identity can be written: [ 1]
(1 )
∫
0
1
(
L
u
)
v
−
u
(
L
v
)
d
x
=
−
p
(
u
′
v
−
u
v
′
)
|
0
1
,
{\displaystyle \ \int _{0}^{1}(Lu)v-u(Lv)\,dx=-p(u'v-uv'){\bigg |}_{0}^{1},}
where
p
=
P
(
x
)
{\displaystyle \ p=P(x)}
q
=
Q
(
x
)
{\displaystyle \ q=Q(x)}
u
=
U
(
x
)
{\displaystyle \ u=U(x)}
v
=
V
(
x
)
{\displaystyle \ v=V(x)}
x
{\displaystyle \ x}
u
{\displaystyle \ u}
v
{\displaystyle \ v}
[
0
,
1
]
{\displaystyle \ [0,1]}
L
{\displaystyle \ L}
Sturm–Liouville differential operator defined by:
(2 )
L
u
=
−
(
p
u
′
)
′
+
q
u
.
{\displaystyle \ Lu=-(pu')'+qu.\ }
Proof
Replace
f
(
x
)
=
p
u
′
{\displaystyle \ f(x)=pu'}
g
(
x
)
=
v
{\displaystyle \ g(x)=v}
a
=
0
{\displaystyle \ a=0}
b
=
1
{\displaystyle \ b=1}
integration by parts
(3 )
∫
a
b
f
′
(
x
)
g
(
x
)
d
x
=
f
(
x
)
g
(
x
)
|
a
b
−
∫
a
b
f
(
x
)
g
′
(
x
)
d
x
,
{\displaystyle \int _{a}^{b}f'(x)\,g(x)\,dx=f(x)\,g(x){\bigg |}_{a}^{b}-\int _{a}^{b}f(x)\,g'(x)\,dx,}
we have:
(4 )
∫
0
1
(
p
u
′
)
′
v
d
x
=
(
p
u
′
)
v
|
0
1
−
∫
0
1
(
p
u
′
)
v
′
d
x
=
p
(
u
′
v
)
|
0
1
−
∫
0
1
(
p
u
′
)
v
′
d
x
.
{\displaystyle \int _{0}^{1}(pu')'v\,dx=(pu')v{\bigg |}_{0}^{1}-\int _{0}^{1}(pu')v'\,dx=p(u'v){\bigg |}_{0}^{1}-\int _{0}^{1}(pu')v'\,dx.}
Replace
f
(
x
)
=
u
{\displaystyle \ f(x)=u}
g
(
x
)
=
p
v
′
{\displaystyle \ g(x)=pv'}
a
=
0
{\displaystyle \ a=0}
b
=
1
{\displaystyle \ b=1}
(' 3 ') again, we have:
∫
0
1
u
′
(
p
v
′
)
d
x
=
u
(
p
v
′
)
|
0
1
−
∫
0
1
u
(
p
v
′
)
′
d
x
{\displaystyle \ \int _{0}^{1}u'(pv')\,dx=u(pv'){\bigg |}_{0}^{1}-\int _{0}^{1}u(pv')'\,dx}
(5 )
−
∫
0
1
(
p
u
′
)
v
′
d
x
=
−
p
(
u
v
′
)
|
0
1
+
∫
0
1
u
(
p
v
′
)
′
d
x
.
{\displaystyle \ -\int _{0}^{1}(pu')v'\,dx=-p(uv'){\bigg |}_{0}^{1}+\int _{0}^{1}u(pv')'\,dx.}
Replace (' 5 ') into (' 4 ') , we get:
∫
0
1
(
p
u
′
)
′
v
d
x
=
p
(
u
′
v
)
|
0
1
−
p
(
u
v
′
)
|
0
1
+
∫
0
1
u
(
p
v
′
)
′
d
x
{\displaystyle \ \int _{0}^{1}(pu')'v\,dx=p(u'v){\bigg |}_{0}^{1}-p(uv'){\bigg |}_{0}^{1}+\int _{0}^{1}u(pv')'\,dx}
(6 )
−
∫
0
1
(
p
u
′
)
′
v
d
x
=
−
p
(
u
′
v
−
u
v
′
)
|
0
1
−
∫
0
1
u
(
p
v
′
)
′
d
x
.
{\displaystyle \ -\int _{0}^{1}(pu')'v\,dx=-p(u'v-uv'){\bigg |}_{0}^{1}-\int _{0}^{1}u(pv')'\,dx.}
From the definition (' 2 ') , we can get:
(7 )
∫
0
1
(
L
u
)
v
d
x
=
∫
0
1
[
−
(
p
u
′
)
′
+
q
u
]
v
d
x
=
−
∫
0
1
(
p
u
′
)
′
v
d
x
+
∫
0
1
u
q
v
d
x
.
{\displaystyle \ \int _{0}^{1}(Lu)v\,dx=\int _{0}^{1}[-(pu')'+qu]v\,dx=-\int _{0}^{1}(pu')'v\,dx+\int _{0}^{1}uqv\,dx.}
Replace (' 6 ') into (' 7 ') , we have:
∫
0
1
(
L
u
)
v
d
x
=
−
p
(
u
′
v
−
u
v
′
)
|
0
1
−
∫
0
1
u
(
p
v
′
)
′
d
x
+
∫
0
1
u
q
v
d
x
{\displaystyle \ \int _{0}^{1}(Lu)v\,dx=-p(u'v-uv'){\bigg |}_{0}^{1}-\int _{0}^{1}u(pv')'\,dx+\int _{0}^{1}uqv\,dx}
(8 )
∫
0
1
(
L
u
)
v
d
x
=
−
p
(
u
′
v
−
u
v
′
)
|
0
1
+
∫
0
1
u
(
L
v
)
d
x
.
{\displaystyle \ \int _{0}^{1}(Lu)v\,dx=-p(u'v-uv'){\bigg |}_{0}^{1}+\int _{0}^{1}u(Lv)\,dx.}
Rearrange terms of (' 8 ') then (' 1 ') is obtained. Q.E.D.
References
^ Boyce, William E. (2001). "Boundary Value Problems and Sturm–Liouville Theory". Elementary Differential Equations and Boundary Value Problems (7th ed. ed.). New York: John Wiley & Sons. p. 630. ISBN 0-471-31999-6 OCLC 64431691 . 
![{\displaystyle \ [0,1]}](/media/api/rest_v1/media/math/render/svg/dd524a5e344fc986442d3b1012c9c75c6a556e28)
![{\displaystyle \ \int _{0}^{1}(Lu)v\,dx=\int _{0}^{1}[-(pu')'+qu]v\,dx=-\int _{0}^{1}(pu')'v\,dx+\int _{0}^{1}uqv\,dx.}](/media/api/rest_v1/media/math/render/svg/4283ca6b5e2b1fca27d7f4c42aabf724e01094dc)
