7.3: Singular points and the method of Frobenius
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 \[2xy'  y =0\] Note that \(x=0\) is a singular point. If we only try to plug in \[ y= \sum_{k=0}^{\infty} a_kx^k,\] we obtain First, \(a_0=0\). Next, the only way to solve for is for for all \(k\). 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 \(x\). Then . So Therefore \(r=1/2\), or in other words \(y=x^{1/2}\). Multiplying by a constant, the general solution for positive \(x\) is \[ y=Cx^{1/2},\] Note that the solution is not even differentiable at \(x=0\). The derivative necessarily must “blow up” at the origin, so much is clear from the differential equation itself. There is only one solution that is differentiable at \9x=0\) and that’s the trivial solution \(y=0\). Not every problem with a singular point has a solution of the form \(y=x^\), of course. But perhaps we can combine the methods. What we will do is to try a solution of the form \[ y = x^r f(x)\] where \(f(x)\) is an analytic function. 
Example 7.3.2 

Suppose that we have the equation and again note that \(x=0\) 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 \(x\). First let us find the derivatives Plugging into our equation we obtain Therefore to 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 \(r=1/2\). 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 \(x^{1}\)) is around as well. It turns out we want to try for another solution of the form which in our case is We would now differentiate this equation, substitute into the differential equation again and solve for \(b_k\). A long computation would ensue and we would obtain some recursion relation for \(b_k\). In fact, the reader can try this to obtain for example the first three terms We would then fix and obtain a solution . Then we write the general solution as . 
7.3.2 The method of Frobenius
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 \(x_0\) is a regular singular point.
Example 7.3.3  

Example 7.3.3: Often, and for the rest of this section, . Consider Write So \(0\) 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 \(0\) is a singular point, but not a regular singular point. 
Theorem 7.3.1 (Method of Frobenius) 
Suppose that \[ p(x)y'' + q(x)y' + r(x)y = 0 \tag{7.3} \] has a regular singular point at \(x=0\), then there exists at least one solution of the form \[ y = x^r \sum_{k=0}^{\infty} a_k x^k.\] A solution of this form is called a Frobeniustype solution. 
The method usually breaks down like this.
 (i) We seek a Frobeniustype 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 Frobeniustype 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 complexvalued function—all the are complex numbers. We obtain our two linearly independent solutions^{5} by taking the real and imaginary parts of .
Note that the main idea is to find at least one Frobeniustype 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.
7.3.3 Bessel functions
An important class of functions that arises commonly in physics are the Bessel functions^{6}. For example, these functions arise as 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 can 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 socalled 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.
7.3.4 Exercises
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).
Contributors
 Jiří Lebl (Oklahoma State University).These pages were supported by NSF grants DMS0900885 and DMS1362337.