SERIES SOLUTIONS; FUCHS’S THEOREM

21. SERIES SOLUTIONS; FUCHS’S THEOREM

We have discussed two examples of differential equations solvable by the Frobenius method (Legendre and Bessel equations). There are many other “named” equations and the corresponding “named” functions which are their solutions. (See a few more examples in Section 22.) All of them have much in common with our two examples and you should not hesitate to look them up and use them without a formal introduction when you run into them on your computer or in a text or reference book. You may discover any or all of the following things about such a new (to you) set of functions: that they are the set of solutions of a differential equation with one or more parameters (like the p in Bessel’s equation); that the values of the functions, their derivatives, their zeros, and many formulas involving them are available in references (tables and computer); that they have orthogonality properties, perhaps with respect to a weight function, and consequently (suitably restricted) functions can be expanded in series of them; that there is a generating function for the set of functions; that there are physical problems whose solutions involve the functions, often in the solution of a partial differential equation; etc.

Now you may wonder whether all differential equations can be solved by the Frobenius method. A general theorem due to Fuchs tells when this method will work; we shall state it for second-order differential equations which are the most important ones in applications. Write the differential equation as

y ′′ + f (x)y ′ + g(x)y = 0.

If xf (x) and x 2 g(x) are expandable in convergent power series ∞ n=0 a n x n , we say that the differential equation (21.1) is regular (or has a nonessential singularity) at the origin. Let us call these the Fuchsian conditions. Fuchs’s theorem says that these conditions are necessary and sufficient for the general solution of (21.1) to consist of either

(1) two Frobenius series, or (2) one solution S 1 (x) which is a Frobenius series, and a second solution which is S 1 (x) ln x + S 2 (x) where S 2 (x) is another Frobenius series.

606 Series Solutions of Differential Equations Chapter 12

Case (2) occurs only when the roots of the indicial equation are equal or differ by an integer, and not always then. [See, for example, equation (11.2) and Problems

11.1 to 11.4, and 11.7 to 11.9.] Note the necessary condition: If the Fuchsian condi- tions are not met, we cannot find the general solution by the method of generalized power series (see Problems 11 to 13). However, the equations most commonly found in applications do meet these conditions.

If the first Frobenius series S 1 (x) happens to break off, or you can easily write its sum in closed form, then the method of “reduction of order” [Chapter 8, Sec- tion 7(e)] gives a way of finding the second solution without using infinite series (see Problems 1 to 4). However, note that our main interest in series solutions is not to solve differential equations this way in general, but to study sets of functions (like Legendre polynomials and Bessel functions) which are solutions of differential equations that occur in applications. So the purpose in using series to solve a few simple differential equations (for which there are easier methods) is to learn how and when the series method works—to watch Fuchs’s theorem in action (see problems).

PROBLEMS, SECTION 21

For Problems 1 to 4, find one (simple) solution of each differential equation by series, and then find the second solution by the “reduction of order” method, Chapter 8, Section 7(e).

1. (x 2 + 1)y ′′ − xy ′ +y=0 2. x 2 y ′′ + (x + 1)y ′ −y=0 3. x 2 y ′′ +x 2 y ′ − 2y = 0

4. (x − 1)y ′′ − xy ′ +y=0 Solve the differential equations in Problems 5 to 10 by the Frobenius method; observe

that you get only one solution. (Note, also, that the two values of s are equal or differ by an integer, and in the latter case the larger s gives the one solution.) Show that the conditions of Fuchs’s theorem are satisfied. Knowing that the second solution is ln x times the solution you have, plus another Frobenius series, find the second solution.

8. xy ′′ + xy ′ − 2y = 0 9. x 2 y ′′ + (x 2 − 3x)y ′ + (4 − 2x)y = 0

10. x 2 (x−1)y ′′ −x(5x−4)y ′ +(9x−6)y = 0 11. For the differential equation in Problem 2, verify that it does not satisfy the Fuchsian

conditions, and that your second solution cannot be expanded in a Frobenius series. 12. Verify that the differential equation x 4 y ′′ + y = 0 is not Fuchsian; that it has the

two independent solutions x sin(1/x) and x cos(1/x); and that these solutions are not expandable in Frobenius series.

13. Verify that the the differential equation in Problem 11.13 is not Fuchsian. Solve it by separation of variables to find the obvious solution y = const. and a second solution in the form of an integral. Show that the second solution is not expandable in a Frobenius series.

Section 22 Hermite Functions; Laguerre Functions; Ladder Operators 607