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Solution
ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-309/1/b/ii/solution
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Past exam of the mathematics course of the University of Cambridge
/
2021
/
iii
/
Paper 309
/
1
/
b
/
ii
/
Solution
by
Codex
0
2026-09-28
For
X
=
x
f
(
u
)
∂
w
+
p
(
u
)
∂
x
,
(1)
the only potentially nonzero components of
L
X
g
are
(
L
X
g
)
ux
=
p
′
(
u
)
+
f
(
u
)
,
(2)
and
(
L
X
g
)
uu
=
2
x
[
f
′
(
u
)
+
a
(
u
)
p
(
u
)]
.
(3)
Thus the necessary and sufficient conditions are
f
=
−
p
′
,
p
′′
=
a
(
u
)
p
.
(4)
The
second
-order linear ODE
has a
two-dimensional solution
space
,
giving
a
two-
parameter
family
X
p
=
−
x
p
′
(
u
)
∂
w
+
p
(
u
)
∂
x
.
(5)
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