Abstract
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 t = 0 and 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.
Highlights
We know the linear Ordinary differential equations (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
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 t = 0 and determined the form of second linearly independent solution
B (t ) = 1 and c (t=) t2 − v2 are analytic at t = 0. This ODE could not be solved by power series method, and it requires the Frobenius method
Summary
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. 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 y′′. B (t ) = 1 and c (t=) t2 − v2 are analytic at t = 0 This ODE could not be solved by power series method, and it requires the Frobenius method. 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
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