# Legendre Polynomial solutions

• h.a.y.l.e.y
In summary, the conversation is about someone seeking help in obtaining an expression for the second solution to a Legendre equation, which may require Sturm Liouville treatment. Another person suggests using Lagrange associate polynomials of the first and second kind to find an independent solution. The original poster has found this information in their lecture notes and is still unsure how to approach the question on their exam. They receive advice to follow Halls' suggestion and are confident in knowing the next steps.
h.a.y.l.e.y
Hi,
I have a problem where I am given the Legendre equation and have been told 1 solution is u(x). It asks me to obtain an expression for the second solution v(x) corresponding to the same value of l.
I think it requires Sturm Liouville treatment but don't have a clue how to begin.

More generally, if you have a known solution, u(x) to a differential equation, Taking
y(x)= u(x)v(x), where v(x) is an unknown function, and plugging into the equation,you get an equation of order one lower for v. In particular, if you know one solution to a second order equation, this will give you a first order equation for an independent solution v(x).

OK so I've skimmed that page and its confirmed what I've got in my lecture notes (albeit in a more complex manner!)
The question remains though, how would I be expected to answer the question, given that this question is only worth 9 marks out of a possible 25 on my exam sheet?!
Surely its asking for something a lot more direct, and I'm still in the dark as to how to begin and what to do...

Take Halls' advice and do what he said.I'm sure u'll get the second indep.solution.

Daniel.

Yes, sorry I was typing that reply whilst Hall's was posted.
Thanks for your help, I think I know where to go from here
x

Can somebody help me in deriving legendre differential function using its generating function

## 1. What are Legendre Polynomial solutions?

Legendre Polynomial solutions are a set of mathematical functions that are used to solve differential equations. These polynomials were first introduced by Adrien-Marie Legendre in the late 18th century and have since been used in various fields of science and engineering.

## 2. What is the significance of Legendre Polynomial solutions?

Legendre Polynomial solutions are significant because they offer a convenient way to solve differential equations, which are often used to model real-world phenomena in physics, engineering, and other fields. They also have many useful properties, such as being orthogonal to each other, which makes them ideal for certain types of mathematical analysis.

## 3. How are Legendre Polynomial solutions calculated?

Legendre Polynomial solutions are typically calculated using a recurrence relation, which involves solving smaller polynomials and using their solutions to find the solutions for larger polynomials. There are also various other methods and techniques used to calculate these solutions, depending on the specific application and desired accuracy.

## 4. What are some common applications of Legendre Polynomial solutions?

Legendre Polynomial solutions have a wide range of applications in science and engineering. They are commonly used in physics to solve problems related to potential energy and wave functions, in signal processing to analyze and filter signals, and in statistics to model data. They are also used in various other fields, such as economics, computer science, and astronomy.

## 5. Are there any limitations to using Legendre Polynomial solutions?

While Legendre Polynomial solutions are useful in many applications, they do have some limitations. They are only applicable to certain types of differential equations and may not always provide the most accurate solutions. Additionally, they can become computationally expensive for higher order polynomials, making them impractical for certain problems.

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