1. Second Order Linear Differential Equations
ECE 6382
Fall 2016
David R. Jackson
Notes 16
Second Order Linear Differential Equations
Notes are from D. R. Wilton, Dept. of ECE

2. Second Order Linear Differential Equations (SOLDEs)
General form of SOLDE:
In linear operator form:
where

3. SOLDEs (cont.) Particular solution: Homogeneous solution:
The general solution is the sum of any particular solution and the homogenous solution.
General solution:

4. SOLDEs (cont.) Comments
Many important differential equations are homogeneous.
The Frobenius method is a general method for finding the homogenous solution (y1 and y2).
For an inhomogeneous equation, the RHS term f usually corresponds to a “source”.
For an inhomogeneous equation, the particular solution can usually be found from the homogenous solution (the method of Green’s functions, for example).
Hence, we focus on the homogenous solution.

5. SOLDEs (cont.)
Un-normalized form:
Normalized form:

6. Frobenius’ Method Our goal is to solve this type of
differential equation (SOLDE) using the
Frobenius method (series solution).
In the Frobenius method we obtain a series solution about some point x = a.
(Often a = 0.)
Ferdinand Georg Frobenius

7. Frobenius’ Method (cont.)
There are three cases:
The point x = a is an ordinary point.
The point x = a is an regular singular point.
The point x = a is an irregular singular point.
Nearest singularity
In all three cases, the Frobenius solution will converge out to the nearest singularity in the coefficient functions.

8. Frobenius’ Method (cont.)
This is the easiest case but not very common.
The Frobenius series is the same as a Taylor series.
This case is difficult and not considered further here.
The Frobenius method does not in general work.
The Frobenius method works for this case.
This is the most common case, and we focus on this case.

9. Frobenius’ Method (cont.)
Example

10. Frobenius’ Method (cont.)

11. Frobenius’ Method (cont.)
Ordinary point:
Taylor series
Taylor series
Both solutions are in the form of Taylor series, and correspond to analytic functions.
This is the easiest case, but unfortunately, it is not so common in practice.

12. Frobenius’ Method (cont.)
Regular singular point:
“Frobenius series”
( is not usually an integer)
(Note: a0 0)
Substitute into DE
Indicial Equation (quadratic equation for : comes from term with smallest exponent)

13. Frobenius’ Method (cont.)
Regular singular point (cont.)
“Frobenius series”
“Frobenius series”
Both solutions are in the form of a Frobenius series.

14. Frobenius’ Method (cont.)
Regular singular point (cont.)
“Frobenius series”
The first solution is in the form of a Frobenius series.
The second solution has a Frobenius series added to a term involving the first solution and a ln function.

15. Frobenius’ Method (cont.)
Regular singular point (cont.)
“Frobenius series”
Case 3a or
Case 3b
The first solution is in the form of a Frobenius series.
The second solution is a Frobenius series or has a Frobenius series added to a term involving the first solution and a ln function (either case is possible).

16. Frobenius’ Method (cont.)
Recipe for the Frobenius method at regular singular point:
Assume a Frobenius series.
Insert Frobenius series into SOLDE.
Obtain indicial equation from the lowest exponent term.
Solve for 1 and 2 from the indicial (quadratic) equation.
Classify the case (1, 2, 3) based on 1 and 2 .
Develop recurrence formula for the coefficients an for y1(x).
Solve for y2(x) by assuming appropriate form*, depending on the case.
*Note: for case 3, try the Frobenius series first (i.e., case 3a). If it doesn't give an independent solution, then use the form for case 3b.

17. Frobenius’ Method (cont.)
Helpful formula that is useful for finding the second solution y2(x):
For cases 2 and 3b, we need to use the following:
(This can be derived after some algebra.)

18. Frobenius’ Method (cont.)
Four Examples
Example 1: Case 1
Example 2: Case 2 (Bessel equation of order 0)
Example 3: Case 3a (Bessel equation of order 1/2)
Example 4: Case 3b (Bessel equation of order 1)
In all cases we choose a = 0 as the expansion point.
In all cases we have a regular singular point.

19. Frobenius’ Method (cont.)
Example 1

20. Frobenius’ Method (cont.)

21. Frobenius’ Method (cont.)
We then have:

22. Frobenius’ Method (cont.)
Example 2
This is the Bessel equation of order zero.
Note: The general Bessel equation of order n is

23. Frobenius’ Method (cont.)

24. Frobenius’ Method (cont.)

25. Frobenius’ Method (cont.)

26. Frobenius’ Method (cont.)
The Bessel function of the first kind, order zero, is defined as:
Choose a0 = 1:

27. Frobenius’ Method (cont.)
Another form:

28. Frobenius’ Method (cont.)

29. Frobenius’ Method (cont.)
If b0  0, this last term would generate y1.

30. Frobenius’ Method (cont.)
Note: N0(x) is often denoted as Y0(x).
Note: The N0 function has a branch cut on the negative real axis.

31. Frobenius’ Method (cont.)
Example 3
This is the Bessel equation of order 1/2.
(This is important in the calculation of the spherical Bessel functions.)
Note: The general Bessel equation of order n is

32. Frobenius’ Method (cont.)

33. Frobenius’ Method (cont.)

34. Frobenius’ Method (cont.)
Hence
Note: We were successful at generating two solutions using only Frobenius series!

35. Frobenius’ Method (cont.)
Note: We don’t need to keep the second term (the sin term) in y2(x), since it is the same as y1(x).
Also, choose the leading constants to be 1.
Hence, we have
Bessel functions of 1/2 order:
Hence,

36. Frobenius’ Method (cont.)
Spherical Bessel functions (of order zero)*:
Hence,
*These are improtant in the solution of the 3D wave equation in spherical coordinates.

37. Frobenius’ Method (cont.)
Example 4
This is the Bessel equation of order 1.
Note: The general Bessel equation of order n is

38. Frobenius’ Method (cont.)

39. Frobenius’ Method (cont.)

40. Frobenius’ Method (cont.)
Hence
The Bessel function of the first kind, order one, is defined as:
Choose a0 = 1/2:

41. Frobenius’ Method (cont.)
We can also do a Frobenius solution to the
general Bessel equation of order n (derivation omitted)*:
where
*In fact, we can even let n  , an arbitrary complex number.

42. Frobenius’ Method (cont.)
Another form:

43. Frobenius’ Method (cont.)
Note: Trying a Frobenius solution with 2 = -1 will fail.
(Try it!)

44. Frobenius’ Method (cont.)

45. Frobenius’ Method (cont.)
Note: N1(x) is often denoted as Y1(x.)
Note: The N1 function has a branch cut on the negative real axis.

46. Frobenius’ Method (cont.)
where

47. Frobenius’ Method (cont.)
where
(Schaum’s Outline Mathematical Handbook, Eq. (24.9))