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
Power Series Solutions:
Substituting Equations 2, 3 and 4 into Equation 1 and rearranging we
arrive at
From Equation 2,
)(
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
11
2
=
=
n
n
dx
x
x
)(
0xY
Power Series Solutions:
First, replacing with ,
Next, replacing with ,
=
m
m
n
nxcxc
2−n
m
n
m
Equation 9
Equation 10
)(
0xY
Power Series Solutions:
Substituting Equations 8 through 11 into Equation 7 we arrive at
The first sum can be written
and we see that the coefficient of , i.e., the constant term, is
, which must be zero, hence
0
x
2
21 c+−
)(
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
0
1=c
0=
i
c
i
1+= in
im 2=
)(
0xY
Power Series Solutions:
From Equations 12 and 13 we see that, for ,
1=i
)(
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
)(
0xJ
)(
0xY
Power Series Solutions:
The higher order functions are defined similarly and as a result
the asymptotic relationships are complementary. For large
)(xYm
x
)(
0xY