Legendre differential equation
What is formation of differential equation?
For any given differential equation, the solution is of the form f(x,y,c1,c2, …….,cn) = 0 where x and y are the variables and c1 , c2 ……. cn are the arbitrary constants. Step 1: Differentiate the given function w.r.t to the independent variable present in the equation.
What are Legendre polynomials used for?
For example, Legendre and Associate Legendre polynomials are widely used in the determination of wave functions of electrons in the orbits of an atom ,  and in the determination of potential functions in the spherically symmetric geometry , etc.
What is hermite differential equation?
Hermite’s Differential Equation is defined as: For is a non-negative integer, i.e., , the solutions of Hermite’s Differential Equation are often referred to as Hermite Polynomials .
What are the two types of differential equation?
We can place all differential equation into two types: ordinary differential equation and partial differential equations.A partial differential equation is a differential equation that involves partial derivatives.An ordinary differential equation is a differential equation that does not involve partial derivatives.
What is linear equation in differential equation?
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form. where , , and are arbitrary differentiable functions that do not need to be linear, and.
What is the general solution of a differential equation?
A solution of a differential equation is an expression for the dependent variable in terms of the independent one(s) which satisfies the relation. The general solution includes all possible solutions and typically includes arbitrary constants (in the case of an ODE) or arbitrary functions (in the case of a PDE.)
Are Legendre polynomials Orthonormal?
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications.
Why are orthogonal polynomials important?
Just as Fourier series provide a convenient method of expanding a periodic function in a series of linearly independent terms, orthogonal polynomials provide a natural way to solve, expand, and interpret solutions to many types of important differential equations.
What is Bessel differential equation?
The Bessel differential equation is the linear second-order ordinary differential equation given by. (1) Equivalently, dividing through by , (2) The solutions to this equation define the Bessel functions and .
How do you solve Chebyshev’s equation?
Then the general solution of the original Chebyshev equation will be given by the formula: y(x)=Ccos(narccosx). In this expression, n may be any real number. But if n is an integer, the given function is the Chebyshev polynomial of the first kind.
What are the types of ordinary differential equations?
Types of Differential EquationsOrdinary Differential Equations.Partial Differential Equations.Linear Differential Equations.Non-linear differential equations.Homogeneous Differential Equations.Non-homogenous Differential Equations.