Differential Equations
for Engineers:
the Essentials
Supplement to Class 25 notes
Partial Differential Equations
Power Series Solutions:
Bessel Function of Second Kind of Order Zero
Bessel Functions: Second Solution
The second fundamental solution to the Bessel equation of order m,
called the Bessel function of the second kind of order m, has
The second solution to Equation 1 can be written
Power Series Solutions:
)(
0xY
The objective of this lecture is to demonstrate that a second solution of
has the form
0
1
2
2=++ y
dx
dy
xdx
yd
Equation 1
Power Series Solutions:
Substituting Equations 2, 3 and 4 into Equation 1 and rearranging we
arrive at
=
−
+
+=
1
1
0
01
ln
n
n
nxnc
x
Jx
dx
dJ
dx
dy
From Equation 2,
Equation 3
2
ln
1
0
0
0
2
0
2
++
++
dx
dJ
x
xJ
dx
dJ
xdx
Jd
)(
0xY
Power Series Solutions:
Now
so Equation 5 becomes, with a little rearrangement,
For Equation 7 to be true for all we can immediately see that
We now need to rearrange Equation 7 until it is in the form
0
1
0
0
2
0
2=++ J
dx
dJ
xdx
Jd
Equation 6
x
0
1=c
Equation 8
)(
0xY
Power Series Solutions:
First, replacing with ,
Next, replacing with ,
2−n
m
=
=++
=
−++=+=
0 1 2
2
2
2
2
2
2
22 )2(2)2(
m m
m
m
m
m
n
n
nxcmcxcmxcn
n
m
12
112
12
1
0)!1(!2
)1(
2)!1(!
)1( −
=−
−
= −
−
=
−
−
=n
nn
n
n
n
nx
nn
x
nndx
dJ
Equation 9
Equation 11
)(
0xY
Power Series Solutions:
Substituting Equations 8 through 11 into Equation 7 we arrive at
The first sum can be written
( )
=+
−
=−=++++
−
−
12
2
2
222
122 0)2(2
)!1(!2
)1(
m
m
mm
n
nn
nxccmcx
nn
)(
0xY
Power Series Solutions:
Substituting Equations 8 through 11 into Equation 7 we arrive at
The first sum involves only even powers. Then we must have
for all the odd powers. Since we find that for odd .
Now let in the first sum and let in the second.
There results
( )
=+
−
=−=+++
−
−
12
222
222 0)2(
)!1(!2
)1(
m
m
mm
n
nn
nxccmx
nn
0
1=c
0=
i
c
i
1+= in
im 2=
)(
0xY
Power Series Solutions:
From Equations 12 and 13 we see that, for ,
=++
0
2
1
2
2
1
2
4
4
3
c
1=i
For
2=i
=++−
06
!2!32
1
46
2
4
cc
)(
0xY
Power Series Solutions:
If we were to continue we would find that
Rather than use the above as the second solution to Equation 1, it is
traditional to employ, instead, a certain linear combination of the
above solution and . That traditional form is
=
−
=n
m
n
n
nmn
c1
22
2
1
)!(2
)1(
)(
0xJ
)(
0xY
Power Series Solutions:
The higher order functions are defined similarly and as a result
the asymptotic relationships are complementary. For large
)(xYm
)
42
cos(
2
)(
−−
mx
x
xJ
m
x
)(
0xY