# Solving ODE Near x=0: Series Solution

• BColl
In summary, the problem is to obtain a solution near x=0 for the equation (x2+1)y''+6xy'+6y=0. The attempt at a solution involves representing the solution in series notation, but there is uncertainty about how to deal with the rational function and adjust the summation to account for it. It is suggested to substitute the series into the original equation, not the one with x2+1 in the denominators.
BColl

## Homework Statement

Obtain solution valid near x=0

## Homework Equations

(x2+1)y''+6xy'+6y=0

## The Attempt at a Solution

y"+6x/(x2+1)y'+6x/(x2+1)=0
In representing the solution in series notation, I'm not sure how deal with the rational function because I know I need to have all of the x terms inside the summation and create powers of x that are all the same to create a single summation. The denominator of a rational function does not seperate, so how would adjust the summation to take this into account?

Substitute the series into the original equation, not the one with x2+1 in the denominators.

(I'm not sure exactly what your question is, but I took a stab at what I thought you meant.)

## 1. How do you solve ODE near x=0 using series solution?

To solve ODEs near x=0 using a series solution, you first need to rewrite the ODE as a power series centered at x=0. This involves expressing each term in the ODE as a polynomial in x, and then finding a recurrence relation between the coefficients of the series. Once you have the series representation of the ODE, you can use techniques such as the method of Frobenius or the method of undetermined coefficients to find a solution.

## 2. What is the advantage of using a series solution for ODEs near x=0?

The advantage of using a series solution for ODEs near x=0 is that it allows for more accurate and precise solutions. Series solutions can capture the behavior of the solution near x=0 with higher precision, making them useful for analyzing systems with small perturbations or near singular points.

## 3. Can the series solution for ODEs near x=0 be used for all types of ODEs?

Not necessarily. The series solution method is most effective for linear ODEs with constant coefficients. For nonlinear or variable coefficient ODEs, the series solution may not converge or may be difficult to obtain. In these cases, other methods such as numerical approximation or perturbation theory may be more suitable.

## 4. Are there any limitations to using series solutions for ODEs near x=0?

Yes, there are some limitations to using series solutions for ODEs near x=0. One limitation is that the series may not converge for all values of x, which can lead to inaccuracies in the solution. Additionally, the series solution may only provide an approximation of the true solution, and may not capture all possible behaviors of the system.

## 5. What are some applications of series solutions for ODEs near x=0?

Series solutions for ODEs near x=0 have various applications in physics, engineering, and other fields. They can be used to model the behavior of physical systems near critical points, such as the behavior of a pendulum near its equilibrium position. Series solutions can also be used to analyze the stability of systems, as well as to solve boundary value problems and initial value problems in a more accurate and precise manner.

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