# Solution of an ODE in series Frobenius method

• I
• Caglar Yildiz
In summary, The series Frobenius method is a technique used to find a solution to a differential equation at a singular point. It involves expanding the solution as a power series and solving for the coefficients using a recurrence relation. It is typically used when the differential equation has a singular point or cannot be solved using other methods. A recurrence relation is used to determine the coefficients in the power series. This method is unique in its ability to solve equations at singular points, but it has limitations and can be complex and time-consuming.
Caglar Yildiz
Hi
I am supposed to find solution of $$xy''+y'+xy=0$$
but i am left with reversing this equation.
i am studying solution of a differential equation by series now and I cannot reverse a series in the form of:
$$J(x)=1-\frac{1}{x^2} +\frac{3x^4}{32} - \frac{5x^6}{576} ...$$

$$\frac{1}{J}=1+\frac{x^2}{2} +\frac{5x^4}{32}+ \frac{23x^6}{576}...$$

General formula of $J(x)$ is $$\sum_{n=0}^{\infty} \frac{(-1)n}{(n!)^2}(\frac{x}{2})^2$$

Thanks for all help!

Why do you need to reverse it?

## 1. What is the series Frobenius method?

The series Frobenius method is a technique used to find a solution to a differential equation of the form y'' + p(x)y' + q(x)y = 0. It involves expanding the solution as a power series and solving for the coefficients using a recurrence relation.

## 2. When is the series Frobenius method used?

The series Frobenius method is typically used when the differential equation has a singular point, such as a point where the coefficients p(x) and q(x) become infinite or when the equation cannot be solved using other methods, such as separation of variables or substitution.

## 3. What is a recurrence relation?

A recurrence relation is a mathematical relationship between the coefficients of a power series. In the series Frobenius method, the recurrence relation is used to determine the values of the coefficients in the power series of the solution.

## 4. How is the series Frobenius method different from other methods of solving ODEs?

The series Frobenius method is unique in that it can be used to find a solution to a differential equation at a singular point. Other methods, such as separation of variables, can only be used for equations with regular points.

## 5. Are there any limitations to the series Frobenius method?

Yes, there are limitations to the series Frobenius method. It can only be used for certain types of differential equations and may not always yield a solution. Additionally, the process of solving for the coefficients in the power series can be time-consuming and complex.

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