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Putnam
1979 Putnam
B4
B4
Part of
1979 Putnam
Problems
(1)
Putnam 1979 B4
Source:
4/8/2022
(a) Find a solution that is not identically zero, of the homogeneous linear differential equation
(
3
x
2
−
x
−
1
)
y
′
′
−
(
9
x
2
+
9
x
−
2
)
y
′
+
(
18
x
+
3
)
y
=
0.
(3x^2-x-1)y''-(9x^2+9x-2)y'+(18x+3)y=0.
(
3
x
2
−
x
−
1
)
y
′′
−
(
9
x
2
+
9
x
−
2
)
y
′
+
(
18
x
+
3
)
y
=
0.
Intelligent guessing of the form of a solution may be helpful. (b) Let
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
be the solution of the nonhomogeneous differential equation
(
3
x
2
−
x
−
1
)
y
′
′
−
(
9
x
2
+
9
x
−
2
)
y
′
+
(
18
x
+
3
)
y
=
6
(
6
x
+
1
)
(3x^2-x-1)y''-(9x^2+9x-2)y'+(18x+3)y=6(6x+1)
(
3
x
2
−
x
−
1
)
y
′′
−
(
9
x
2
+
9
x
−
2
)
y
′
+
(
18
x
+
3
)
y
=
6
(
6
x
+
1
)
that has
f
(
0
)
=
1
f(0)=1
f
(
0
)
=
1
and
(
f
(
−
1
)
−
2
)
(
f
(
1
)
−
6
)
=
1.
(f(-1)-2)(f(1)-6)=1.
(
f
(
−
1
)
−
2
)
(
f
(
1
)
−
6
)
=
1.
Find integers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
(
f
(
−
2
)
−
a
)
(
f
(
2
)
−
b
)
=
c
.
(f(-2)-a)(f(2)-b)=c.
(
f
(
−
2
)
−
a
)
(
f
(
2
)
−
b
)
=
c
.
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