# Solve Frobenius Series Homework for 2 Independent Solutions

• Ted123
In summary, the question asks to find 2 independent solutions for the given equation using Frobenius series in x. The solution involves substituting y = \sum_{n=0}^{\infty} a_n x^{n + \sigma} and solving for the indicial equation. The resulting recurrence relation is used to find the two possible values of \sigma, which in turn are used to find the two independent solutions. The question also mentions that x=0 is a regular singular point, but it is not clear if this has any significance in the solution process.
Ted123

## Homework Statement

The origin is a regular singular point of the equation $$2x^2 y'' + xy' - (x+1)y =0.$$ Find 2 independent solutions which are Frobenius series in x.

## The Attempt at a Solution

Substituting $$y = \sum_{n=0}^{\infty} a_n x^{n + \sigma}$$ eventually gives $$(2\sigma(\sigma - 1) +\sigma -1 )a_0 x^{\sigma} + \sum_{n=0}^{\infty} \left[ (2(\sigma + n)(\sigma + n+1) + \sigma + n ) a_{n+1} - a_n \right] x^{n+\sigma + 1} = 0.$$

Equating the series to 0 term-by-term gives the indicial equation $$2\sigma (\sigma -1) + \sigma -1 = 0 \Rightarrow (2\sigma +1)(\sigma -1) = 0 \Rightarrow \sigma = -\frac{1}{2},\; \sigma = 1.$$

We get the recurrence relation $$a_{n+1} = \frac{a_n}{2(\sigma + n)(\sigma + n +1) + \sigma + n}.$$

This is what I'm struggling to solve...

Last edited:
Don't you have the solution now? You know there are two possible values of sigma. For each sigma you have a slightly different recursion, leading to a slightly different series. In general, you have $$\displaystyle a_1 = \frac{a_0}{2 \sigma (\sigma +1) + \sigma} \:,$$
$$\displaystyle a_2 = a_0 \prod_{i=0}^{1} \frac{1}{2(\sigma+i)(\sigma+i+1) + \sigma+i},$$
etc.

RGV

So the 2 independent solutions, for $$\sigma =1, -\frac{1}{2}$$ is:

$$\displaystyle a_n = a_0 \prod_{i=0}^{n-1} \frac{1}{2(\sigma+i)(\sigma+i+1) + \sigma+i}\;?$$

Is there any significance to the question mentioning that x=0 is a regular singular point?

Last edited:

## What is a Frobenius series?

A Frobenius series is a type of power series that is used to solve differential equations with singular points. It was first introduced by German mathematician Ferdinand Georg Frobenius in the late 19th century.

## What is the significance of having 2 independent solutions in a Frobenius series?

Having 2 independent solutions in a Frobenius series is significant because it allows us to find a general solution to a differential equation with a singular point. This is important because it provides a more complete understanding of the behavior of the system being modeled.

## How do you solve a Frobenius series homework problem?

To solve a Frobenius series homework problem, you first need to determine the indicial equation, which will give you the values of the two exponents that will be used in the series. Then, you can use the recurrence relation to find the coefficients of the series and plug them into the general solution formula. Finally, you can use initial conditions to determine the specific solution to the problem.

## What are some common mistakes when solving Frobenius series homework?

One common mistake when solving Frobenius series homework is forgetting to check for convergence of the series. Another mistake is using the wrong values for the exponents in the series. It is also important to be careful with algebraic manipulations and to double check all calculations.

## What are some tips for improving understanding and accuracy when solving Frobenius series homework?

Some tips for improving understanding and accuracy when solving Frobenius series homework include practicing a lot of examples, reviewing the theory behind Frobenius series, and seeking help from a tutor or instructor if needed. It is also helpful to break down the problem into smaller steps and to check your work carefully.

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