We know the linear ODEs with constant coefficients can be solved by functions known from calculus.
If a linear ordinary differential equation has variable coefficients, like Legendre’s and Bessel’s ODEs, it must be solved by other methods.
The power series method is a very effective method for solving the ODEs which have coefficient variable. It gives solution in the form of power series.
A power series is an infinite series of the form
in which t is a variable. are constants, which are the coefficients of the series. is the center of the series. In particular, if , we get a power series in power of t
If a differential equation is given in the form
Here and are analytic functions at .
This equation can be written in standard form below
If and are analytic at , then the solution of the equation will be analytic at , which can be represented in the form
However if either or are not analytic at , in other words we have a singular point at . Then solution cannot be represented in the series, so we must go to power series expanded method which is called Frobenius method.
The Frobenius method enables us to solve such types of differential equations for example, Bessel’s equation
of the form of “Equation (4)”. Here and are analytic at . This ODE could not be solved by power series method, and it requires the Frobenius method.
P. Haarsa and S. Pothat have considered such types of ODEs, but they acquired exclusively the first solution besides general solution. Similarly, Anil Hakim Syofra, Rika Permatasari and Lily Adriani Nazara attained the form of the second solution in real case of the mentioned equations in their research paper.
1) We will study how we can solve second order ODEs at a singular point.
2) Discuss the real and complex cases of the solution with examples.
2. Regular and Singular Point
A regular point of
is a point in which the coefficients and are analytic functions. Likewise, a regular point of the ordinary differential equation
is in which and are analytic and (divide by we get the standard form). So the power series method can be applied. If is not a regular point, it is called a singular point  .
3. Frobenius Method
If is a singular point of the ordinary differential “Equation (4)”, then it has at least one solution of the form
in which k may be any (real or complex) number .
The second-order differential “Equation (4)” also has a second solution which may be similar to solution one with a different k and different coefficients, or may have a logarithmic term. Solutions one and two are linearly independent  .
4. Indicial Equation
Now we shall discuss the method of Frobenius for solving “Equation (4)” at a singular point . Multiply “Equation (4)” by , we get
Now and are expanding in power series,
Differentiating “Equation (6)” term by term, finding
By putting the valves of , and into “Equation (7)”, we readily obtain
equating the sum of the coefficients of each power of to zero.
This gives a structure of equations with the unknown coefficients .
The corresponding equation to the power is
Since by assumption that the above expression must be zero. This gives
This equation is important and is known as indicial equation of the ordinary differential “Equation (4)”. It plays the role as follows:
Method of Frobenius gives a basis of solution. One solution of the given ODE will be of the form of “Equation (6)”, where k is a root of “Equation (10)”. The second one will be of the form specified by the indicial equation  .
Let the roots of indicial equation be and . Then we have：
The Real Case: suppose and are real and . There are three cases as follows:
Case 1: Distinct roots not different by an integer
If and are such that is not an integer. Then we have
with coefficients obtained successively from “Equation (9)” in which and respectively   .
We solve the ODE
Its standard form is
Substitute and its derivatives and into the above equation, we get
We have the axillary equation
0.5 and −1 are the roots. Hence tow solutions for all positive t is and  .
Case 2: Double roots ( )
A basis is
In case tow we must have algorithm, where in case three we may or may not. Therefore the second solution is
We solve the ODE
Writing “Equation (15)” in standard form of “Equation (4)”
We see that it satisfies the mentioned condition, by putting “Equation (6)” and its derivatives into “Equation (15)”, we get
The slightest power is , occurring in the second and the fourth series; by associating the sum of its factors to zero, we have
Therefore this indicial equation has the paired roots  .
By addition the value of in (16) and compare the sum of the power to zero, we attain
. Hence . And by selecting , we get the answer
For second independent solution, we apply the method of decrease of order, replacing and its derivatives into equation. We have , the factor of in (15) in standard form.
By partial fractions,
These functions are linearly independent and thus form a basis on the interval (as well as on )   .
Case 3: is a positive integer
The tow solutions are
We solve the following ODE
Replacing “Equation (6)” and “Equation (8)”, in “Equation (19)”, we get
By taking , t and t exclusive the summations and bring together all terms with power and simplify algebraically,
We set in the first series and in the second series, thus . Then
Here the lowest power is and provides the indicial equation
and are the roots, they are different by an integer.
From (20) with , we have
From this we get the recurrence relation
For , we get first solution as .
Applying the method of reduction of order, we replace , and into the ODE (19), we have
Drops out tu. Division by t and simplification gives
From partial fractions and integrating, we obtain
By taking exponents and integrating again, we get
These two solutions are linearly independent, and has logarithmic term  .
The Complex Case: If the roots of the indicial equation are distinct complex
Numbers, k and say, then the solution is a complex-valued and is of the form
in which the coefficients may be complex. and are the real and imaginary parts of , respectively and are linearly independent .
Replacing “Equation (6)” and “Equation (8)”, in “Equation (22)”, we obtain
We know the initial equation is
Its roots Thus, there is a complex-valued solution.
Since the index of the sums has unlike starting points, we separate the cases and for getting the following:
Then case implies that , as standard, is arbitrary but nonzero.
Substituting (the case will give same result) and into
the cases and above, gives and . The general recurrence relation is
It follows that
Suppose , Therefore, we can write and
By using Euler’s formula, the real and imaginary parts are
There is at least one Frobenius solution, in each case. When the roots of initial equation are real, there is a Frobenius solution for the larger of the two roots. If is a Frobenius solution and there is not a second solution, then a second independent solution is the sum of a logarithmic expression and a Frobenius series. If there are complex roots, the first and second solutions will be of the real and imaginary parts of .
When is a singular point of the second-order ordinary differential equation
in which and are not analytic in , the Frobenius method will apply and it has at least one solution which can be represented in the form
The above ordinary differential equation also has a second solution such that they are linearly independent.
Its form will be specified by “Equation (6)” in the following cases.
The real case: If the roots of the initial equation are real, then there are the following cases:
Case 1: is not an integer
Case 2: double roots
Case 3: roots differing by an integer
The complex case: When the roots of initial equation are distinctly complex, solution is of the form
in which the coefficients may be complex. and are the real and imaginary parts of , respectively.
 Syofra, A.H., Permatasari, R. and Nazara, A. (2016) The Frobenius Method for Solving Ordinary Differential Equation with Coefficient Variable. IJSR, 5, 2233-2235.