Secondorder linear ordinary differential equations

16
Series solutions of ordinary
differential equations
In the previous chapter the solution of both homogeneous and non-homogeneous
linear ordinary differential equations (ODEs) of order ≥ 2 was discussed. In particular we developed methods for solving some equations in which the coefficients
were not constant but functions of the independent variable x. In each case we
were able to write the solutions to such equations in terms of elementary functions, or as integrals. In general, however, the solutions of equations with variable
coefficients cannot be written in this way, and we must consider alternative
approaches.
In this chapter we discuss a method for obtaining solutions to linear ODEs
in the form of convergent series. Such series can be evaluated numerically, and
those occurring most commonly are named and tabulated. There is in fact no
distinct borderline between this and the previous chapter, since solutions in terms
of elementary functions may equally well be written as convergent series (i.e. the
relevant Taylor series). Indeed, it is partly because some series occur so frequently
that they are given special names such as sin x, cos x or exp x.
Since we shall be concerned principally with second-order linear ODEs in this
chapter, we begin with a discussion of these equations, and obtain some general
results that will prove useful when we come to discuss series solutions.
16.1 Second-order linear ordinary differential equations
Any homogeneous second-order linear ODE can be written in the form
y + p(x)y + q(x)y = 0,
(16.1)
where y = dy/dx and p(x) and q(x) are given functions of x. From the previous
chapter, we recall that the most general form of the solution to (16.1) is
y(x) = c1 y1 (x) + c2 y2 (x),
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(16.2)
SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
where y1 (x) and y2 (x) are linearly independent solutions of (16.1), and c1 and c2
are constants that are fixed by the boundary conditions (if supplied).
A full discussion of the linear independence of sets of functions was given
at the beginning of the previous chapter, but for just two functions y1 and y2
to be linearly independent we simply require that y2 is not a multiple of y1 .
Equivalently, y1 and y2 must be such that the equation
c1 y1 (x) + c2 y2 (x) = 0
is only satisfied for c1 = c2 = 0. Therefore the linear independence of y1 (x) and
y2 (x) can usually be deduced by inspection but in any case can always be verified
by the evaluation of the Wronskian of the two solutions,
y y2 = y1 y2 − y2 y1 .
W (x) = 1
(16.3)
y1 y2 If W (x) = 0 anywhere in a given interval then y1 and y2 are linearly independent
in that interval.
An alternative expression for W (x), of which we will make use later, may be
derived by differentiating (16.3) with respect to x to give
W = y1 y2 + y1 y2 − y2 y1 − y2 y1 = y1 y2 − y1 y2 .
Since both y1 and y2 satisfy (16.1), we may substitute for y1 and y2 to obtain
W = −y1 (py2 + qy2 ) + (py1 + qy1 )y2 = −p(y1 y2 − y1 y2 ) = −pW .
Integrating, we find
W (x) = C exp −
x
p(u) du ,
(16.4)
where C is a constant. We note further that in the special case p(x) ≡ 0 we obtain
W = constant.
The functions y1 = sin x and y2 = cos x are both solutions of the equation y + y =
0. Evaluate the Wronskian of these two solutions, and hence show that they are linearly
independent.
The Wronskian of y1 and y2 is given by
W = y1 y2 − y2 y1 = − sin2 x − cos2 x = −1.
Since W = 0 the two solutions are linearly independent. We also note that y + y = 0 is
a special case of (16.1) with p(x) = 0. We therefore expect, from (16.4), that W will be a
constant, as is indeed the case. From the previous chapter we recall that, once we have obtained the general
solution to the homogeneous second-order ODE (16.1) in the form (16.2), the
general solution to the inhomogeneous equation
y + p(x)y + q(x)y = f(x)
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(16.5)
16.1 SECOND-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS
can be written as the sum of the solution to the homogeneous equation yc (x)
(the complementary function) and any function yp (x) (the particular integral) that
satisfies (16.5) and is linearly independent of yc (x). We have therefore
y(x) = c1 y1 (x) + c2 y2 (x) + yp (x).
(16.6)
General methods for obtaining yp , that are applicable to equations with variable
coefficients, such as the variation of parameters or Green’s functions, were discussed in the previous chapter. An alternative description of the Green’s function
method for solving inhomogeneous equations is given in the next chapter. For the
present, however, we will restrict our attention to the solutions of homogeneous
ODEs in the form of convergent series.
16.1.1 Ordinary and singular points of an ODE
So far we have implicitly assumed that y(x) is a real function of a real variable
x. However, this is not always the case, and in the remainder of this chapter we
broaden our discussion by generalising to a complex function y(z) of a complex
variable z.
Let us therefore consider the second-order linear homogeneous ODE
y + p(z)y + q(z) = 0,
(16.7)
where now y = dy/dz; this is a straightforward generalisation of (16.1). A full
discussion of complex functions and differentiation with respect to a complex
variable z is given in chapter 24, but for the purposes of the present chapter we
need not concern ourselves with many of the subtleties that exist. In particular,
we may treat differentiation with respect to z in a way analogous to ordinary
differentiation with respect to a real variable x.
In (16.7), if, at some point z = z0 , the functions p(z) and q(z) are finite and can
be expressed as complex power series (see section 4.5), i.e.
p(z) =
∞
pn (z − z0 )n ,
q(z) =
n=0
∞
qn (z − z0 )n ,
n=0
then p(z) and q(z) are said to be analytic at z = z0 , and this point is called an
ordinary point of the ODE. If, however, p(z) or q(z), or both, diverge at z = z0
then it is called a singular point of the ODE.
Even if an ODE is singular at a given point z = z0 , it may still possess a
non-singular (finite) solution at that point. In fact the necessary and sufficient
condition§ for such a solution to exist is that (z − z0 )p(z) and (z − z0 )2 q(z) are
both analytic at z = z0 . Singular points that have this property are called regular
§
See, for example, H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics, 3rd edn (Cambridge: Cambridge University Press, 1966), p. 479.
533
SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS
singular points, whereas any singular point not satisfying both these criteria is
termed an irregular or essential singularity.
Legendre’s equation has the form
(1 − z 2 )y − 2zy + ( + 1)y = 0,
(16.8)
where is a constant. Show that z = 0 is an ordinary point and z = ±1 are regular singular
points of this equation.
Firstly, divide through by 1 − z 2 to put the equation into our standard form (16.7):
2z
( + 1)
y +
y = 0.
1 − z2
1 − z2
Comparing this with (16.7), we identify p(z) and q(z) as
y −
p(z) =
−2z
−2z
=
,
1 − z2
(1 + z)(1 − z)
q(z) =
( + 1)
( + 1)
=
.
1 − z2
(1 + z)(1 − z)
By inspection, p(z) and q(z) are analytic at z = 0, which is therefore an ordinary point,
but both diverge for z = ±1, which are thus singular points. However, at z = 1 we see
that both (z − 1)p(z) and (z − 1)2 q(z) are analytic and hence z = 1 is a regular singular
point. Similarly, at z = −1 both (z + 1)p(z) and (z + 1)2 q(z) are analytic, and it too is a
regular singular point. So far we have assumed that z0 is finite. However, we may sometimes wish to
determine the nature of the point |z| → ∞. This may be achieved straightforwardly
by substituting w = 1/z into the equation and investigating the behaviour at
w = 0.
Show that Legendre’s equation has a regular singularity at |z| → ∞.
Letting w = 1/z, the derivatives with respect to z become
dy
dy
dy dw
1 dy
=
=− 2
= −w 2
,
dz
dw dz
z dw
dw
2
dy
dy
dw d
d2 y
dy
d2 y
dy
=
= −w 2 −2w
− w2 2 = w3 2
+w 2 .
2
dz
dz dw dz
dw
dw
dw
dw
If we substitute these derivatives into Legendre’s equation (16.8) we obtain
dy
1
1
dy
d2 y
1 − 2 w3 2
+ w 2 + 2 w2
+ ( + 1)y = 0,
w
dw
dw
w dw
which simplifies to give
d2 y
dy
+ 2w 3
+ ( + 1)y = 0.
dw 2
dw
2
2
Dividing through by w (w − 1) to put the equation into standard form, and comparing
with (16.7), we identify p(w) and q(w) as
w 2 (w 2 − 1)
p(w) =
2w
,
w2 − 1
q(w) =
( + 1)
.
w 2 (w 2 − 1)
At w = 0, p(w) is analytic but q(w) diverges, and so the point |z| → ∞ is a singular point
of Legendre’s equation. However, since wp and w 2 q are both analytic at w = 0, |z| → ∞
is a regular singular point. 534