# Solving an ODE with Legendre Polynomials

• I
• CrosisBH
In summary, by manipulating the given ODE using trigonometric identities and introducing a new variable, we can see that the solution is in the form of Legendre Polynomials. This method can be applied to other ODEs to find their solutions in the form of Legendre Polynomials as well.
CrosisBH
TL;DR Summary
Sol
From Griffiths E&M 4th edition. He went over solving a PDE using separation of variables. It got to this ODE
$$\frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right)= -l(l+1)\sin \theta \Theta$$
Griffths states that this ODE has the solution
$$\Theta = P_l(\cos\theta)$$
Where $$P_l = \frac{1}{2^l !} \frac{d^l}{dx^l} (x^2 - 1)^l$$ is a Legendre Polynomial. I was curious to see how this generalizes. I found the definition of the Legendre's Polynomials is
[source]

I have trouble seeing how this is the form of the ODE above. I've tried playing with it but I can't get it into the form where it makes sense that the Legendre Polynomials are the solution. I'm also curious of more examples of ODEs that can be manipulated into this form. Thank you!

Delta2
Begin by deviding both side of
$$\frac{d}{d\theta}\bigg(\sin\theta\frac{d\Theta}{d\theta}\bigg) = - l(l+1)\sin\theta\,\Theta$$
by ##\sin\theta## to get
$$\frac{1}{\sin\theta}\frac{d}{d\theta}\bigg(\sin\theta\frac{d\Theta}{d\theta}\bigg) + l(l+1)\Theta = 0.$$
Now, introduce the new variable ##x=-\cos\theta## and thus
$$\frac{d}{d\theta} = \sin\theta\frac{d}{dx}.$$
Therefore
$$\frac{d}{dx}\bigg(\sin^2\theta\frac{d\Theta}{dx}\bigg) + l(l+1)\Theta = 0.$$
Lastly, notice that ##x^2 = \cos^2\theta## together with the trigonometric identity ##\sin^2\theta = 1-\cos^2\theta##. Thus,
$$\frac{d}{dx}\bigg((1-x^2)\frac{d\Theta}{dx}\bigg) + l(l+1)\Theta = 0$$
which exactly is Legendre's differential equation.

CrosisBH

## 1. What is an ODE?

An ODE, or ordinary differential equation, is a mathematical equation that describes the relationship between a function and its derivatives. It is commonly used to model physical systems in science and engineering.

## 2. What are Legendre polynomials?

Legendre polynomials are a set of orthogonal polynomials that are commonly used to solve differential equations. They are named after French mathematician Adrien-Marie Legendre and are often used to solve problems in physics and engineering.

## 3. How are Legendre polynomials used to solve ODEs?

Legendre polynomials can be used to solve ODEs by representing the solution as a series of these polynomials. This allows for a more efficient and accurate solution compared to other methods.

## 4. What are the advantages of using Legendre polynomials to solve ODEs?

Using Legendre polynomials to solve ODEs has several advantages, including their orthogonality which allows for accurate and efficient solutions. They also have a wide range of applications and are well-suited for solving problems with boundary conditions.

## 5. Are there any limitations to using Legendre polynomials to solve ODEs?

While Legendre polynomials are a powerful tool for solving ODEs, they do have some limitations. They are not suitable for all types of differential equations and may not always provide the most accurate solution. Other methods, such as numerical methods, may be more appropriate for certain types of ODEs.

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