Frobenius Method for Solving Second-Order Ordinary Differential Equations

As we know that the power series method is a very effective method for solving the Ordinary differential equations (ODEs) which have variable coefficient, so in this paper we have studied how to solve second-order ordinary differential equation with variable coefficient at a singular point determined the form of second linearly independent solution. Based on the roots of initial equation there are real and complex cases. When the roots of initial equation are real then there are three kinds of second linearly independent solutions. If the roots of the initial equation are distinct complex numbers, then the solution is complex-valued.


Introduction
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   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 are analytic at 0 t = . 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.

Frobenius Method
If 0 t = is a singular point of the ordinary differential "Equation (4) in which k may be any (real or complex) number [3].
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 [2] [4].

Indicial Equation
Now we shall discuss the method of Frobenius for solving "Equation (4)" at a singular point 0 t = . Multiply "Equation By putting the valves of ( ) equating the sum of the coefficients of each power of  to zero. This gives a structure of equations with the unknown coefficients r a .
The corresponding equation to the power k t is Since by assumption that 0 0 a ≠ the above expression must be zero. This 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 [ In case tow we must have algorithm, where in case three we may or may not. Therefore the second solution is Example 2: 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    Replacing "Equation (6) Here the lowest power is

Result
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

Conclusions
When 0 t = is a singular point of the second-order ordinary differential equation ( ) ( ) The above ordinary differential equation also has a second solution such that they are linearly independent.
Its form will be specified by "Equation (6)