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Past exam of the mathematics course of the University of Cambridge
/
2024
/
ii
/
Paper 2
/
7D
/
a
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Mathematics course of the University of Cambridge
Past exam of the mathematics course of the University of Cambridge
2024
ii
Paper 2
7D
Created
2026-09-23
Updated
2026-09-24
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Solution
a
Solution
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a
Substituting
y
(
z
)
=
∫
γ
e
z
t
f
(
t
)
d
t
,
y
(
3
)
(
z
)
=
∫
γ
e
z
t
t
3
f
(
t
)
d
t
,
(1)
and using
z
e
z
t
=
∂
t
e
z
t
gives
z
y
(
3
)
+
2
y
=
[
e
z
t
t
3
f
(
t
)
]
∂
γ
+
∫
γ
e
z
t
{
2
f
(
t
)
−
(
t
3
f
(
t
)
)
′
}
d
t
.
(2)
It is therefore enough to require
(
t
3
f
)
′
=
2
f
(3)
and choose
γ
so that the
integral
converges and
e
z
t
t
3
f
(
t
)
vanishes at its endpoints. Solving the
amplitude equation
,
f
f
′
=
t
3
2
−
t
3
,
f
(
t
)
=
C
t
−
3
e
−
1/
t
2
.
(4)
Ancestors
(10)
7D
Paper 2
ii
2024
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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