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Note: 1 or 1.5 lectures, §8.4 and §8.5 in [EP], §5.4–§5.7 in [BD]

While behavior of ODEs at singular points is more complicated, certain singular points are not especially diﬃcult to solve. Let us look at some examples before giving a general method. We may be lucky and obtain a power series solution using the method of the previous section, but in general we may have to try other things.

Example 7.3.1: Let us ﬁrst look at a simple ﬁrst order equation

Note that is a singular point. If we only try to plug in

we obtain

First, . Next, the only way to solve for is for for all . Therefore we only get the trivial solution . We need a nonzero solution to get the general solution.

Let us try for some real number . Consequently our solution—if we can ﬁnd one—may only make sense for positive . Then . So

Therefore , or in other words . Multiplying by a constant, the general solution for positive is

If then the derivative of the solution “blows up” at (the singular point). There is only one solution that is diﬀerentiable at and that’s the trivial solution .

Not every problem with a singular point has a solution of the form , of course. But perhaps we can combine the methods. What we will do is to try a solution of the form

where is an analytic function.

Example 7.3.2: Suppose that we have the equation

and again note that is a singular point.

Let us try

where is a real number, not necessarily an integer. Again if such a solution exists, it may only exist for positive . First let us ﬁnd the derivatives

Plugging into our equation we obtainTo have a solution we must ﬁrst have . Supposing that we obtain

This equation is called the indicial equation. This particular indicial equation has a double root at .

OK, so we know what has to be. That knowledge we obtained simply by looking at the coeﬃcient of . All other coeﬃcients of also have to be zero so

If we plug in and solve for we get

Let us set . Then

Extrapolating, we notice that

In other words,

That was lucky! In general, we will not be able to write the series in terms of elementary functions.

We have one solution, let us call it . But what about a second solution? If we want a general solution, we need two linearly independent solutions. Picking to be a diﬀerent constant only gets us a constant multiple of , and we do not have any other to try; we only have one solution to the indicial equation. Well, there are powers of ﬂoating around and we are taking derivatives, perhaps the logarithm (the antiderivative of ) is around as well. It turns out we want to try for another solution of the form

which in our case is

We now diﬀerentiate this equation, substitute into the diﬀerential equation and solve for . A long computation ensues and we obtain some recursion relation for . The reader can (and should) try this to obtain for example the ﬁrst three terms

We then ﬁx and obtain a solution . Then we write the general solution as .

Before giving the general method, let us clarify when the method applies. Let

be an ODE. As before, if , then is a singular point. If, furthermore, the limits

both exist and are ﬁnite, then we say that is a regular singular point.

Example 7.3.3: Often, and for the rest of this section, . Consider

Write

So is a regular singular point.On the other hand if we make the slight change

then

Here DNE stands for does not exist. The point is a singular point, but not a regular singular point.

Let us now discuss the general Method of Frobenius^{4}. Let us only consider the method at the point for simplicity. The main idea is the following theorem.

Theorem 7.3.1 (Method of Frobenius). Suppose that

(7.3) |

has a regular singular point at , then there exists at least one solution of the form

A solution of this form is called a Frobenius-type solution.

The method usually breaks down like this.

- (i)
- We seek a Frobenius-type solution of the form
We plug this into equation (7.3). We collect terms and write everything as a single series.

- (ii)
- The obtained series must be zero. Setting the ﬁrst coeﬃcient (usually the coeﬃcient of ) in the series to zero we obtain the indicial equation, which is a quadratic polynomial in .
- (iii)
- If the indicial equation has two real roots and such that is not an integer, then we have two linearly independent Frobenius-type solutions. Using the ﬁrst root, we plug in
and we solve for all to obtain the ﬁrst solution. Then using the second root, we plug in

and solve for all to obtain the second solution.

- (iv)
- If the indicial equation has a doubled root , then there we ﬁnd one solution
and then we obtain a new solution by plugging

into equation (7.3) and solving for the constants .

- (v)
- If the indicial equation has two real roots such that is an integer, then one solution is
and the second linearly independent solution is of the form

where we plug into (7.3) and solve for the constants and .

- (vi)
- Finally, if the indicial equation has complex roots, then solving for in the solution
results in a complex-valued function—all the are complex numbers. We obtain our two linearly independent solutions

^{5}by taking the real and imaginary parts of .

The main idea is to ﬁnd at least one Frobenius-type solution. If we are lucky and ﬁnd two, we are done. If we only get one, we either use the ideas above or even a diﬀerent method such as reduction of order (Exercise 2.1.8) to obtain a second solution.

An important class of functions that arises commonly in physics are the Bessel functions^{6}. For example, these functions appear when solving the wave equation in two and three dimensions. First we have Bessel’s equation of order :

We allow to be any number, not just an integer, although integers and multiples of are most important in applications.

When we plug

into Bessel’s equation of order we obtain the indicial equation

Therefore we obtain two roots and . If is not an integer following the method of Frobenius and setting , we obtain linearly independent solutions of the form

Exercise 7.3.1: a) Verify that the indicial equation of Bessel’s equation of order is . b) Suppose that is not an integer. Carry out the computation to obtain the solutions and above.

Bessel functions will be convenient constant multiples of and . First we must deﬁne the gamma function

Notice that . The gamma function also has a wonderful property

From this property, one can show that when is an integer, so the gamma function is a continuous version of the factorial. We compute:

We deﬁne the Bessel functions of the ﬁrst kind of order and as

As these are constant multiples of the solutions we found above, these are both solutions to Bessel’s equation of order . The constants are picked for convenience.When is not an integer, and are linearly independent. When is an integer we obtain

In this case it turns out that

and so we do not obtain a second linearly independent solution. The other solution is the so-called Bessel function of second kind. These make sense only for integer orders and are deﬁned as limits of linear combinations of and as approaches in the following way:

As each linear combination of and is a solution to Bessel’s equation of order , then as we take the limit as goes to , is a solution to Bessel’s equation of order . It also turns out that and are linearly independent. Therefore when is an integer, we have the general solution to Bessel’s equation of order

for arbitrary constants and . Note that goes to negative inﬁnity at . Many mathematical software packages have these functions and deﬁned, so they can be used just like say and . In fact, they have some similar properties. For example, is a derivative of , and in general the derivative of can be written as a linear combination of and . Furthermore, these functions oscillate, although they are not periodic. See Figure 7.4 for graphs of Bessel functions.

Example 7.3.4: Other equations can sometimes be solved in terms of the Bessel functions. For example, given a positive constant ,

can be changed to . Then changing variables we obtain via chain rule the equation in and :

which can be recognized as Bessel’s equation of order 0. Therefore the general solution is , or in terms of :

This equation comes up for example when ﬁnding fundamental modes of vibration of a circular drum, but we digress.

Exercise 7.3.8: In the following equations classify the point as ordinary, regular singular, or singular but not regular singular.

- a)
- b)
- c)
- d)
- e)

Exercise 7.3.101: In the following equations classify the point as ordinary, regular singular, or singular but not regular singular.

- a)
- b)
- c)
- d)
- e)

^{4}Named after the German mathematician Ferdinand Georg Frobenius (1849–1917).

^{5}See Joseph L. Neuringera, The Frobenius method for complex roots of the indicial equation, International Journal of Mathematical Education in Science and Technology, Volume 9, Issue 1, 1978, 71–77.

^{6}Named after the German astronomer and mathematician Friedrich Wilhelm Bessel (1784–1846).