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The Legendre differential equation is the second-order ordinary differential equation (1-x^2)(d^2y)/(dx^2)-2x(dy)/(dx)+l(l+1)y=0, (1) which can be rewritten ...

The associated Legendre differential equation is a generalization of the Legendre differential equation given by d/(dx)[(1-x^2)(dy)/(dx)]+[l(l+1)-(m^2)/(1-x^2)]y=0, (1) which ...

An equation is a mathematical expression stating that two or more quantities are the same as one another, also called an equality, formula, or identity.

The word differential has several related meaning in mathematics. In the most common context, it means "related to derivatives." So, for example, the portion of calculus ...

The second-order ordinary differential equation (1-x^2)y^('')-2(mu+1)xy^'+(nu-mu)(nu+mu+1)y=0 (1) sometimes called the hyperspherical differential equation (Iyanaga and ...

An ordinary differential equation (frequently called an "ODE," "diff eq," or "diffy Q") is an equality involving a function and its derivatives. An ODE of order n is an ...

A differential equation is an equation that involves the derivatives of a function as well as the function itself. If partial derivatives are involved, the equation is called ...

On the surface of a sphere, attempt separation of variables in spherical coordinates by writing F(theta,phi)=Theta(theta)Phi(phi), (1) then the Helmholtz differential ...

A functional differential equation is a differential equation in which the derivative y^'(t) of an unknown function y has a value at t that is related to y as a function of ...

(1) or (2) The solutions are Jacobi polynomials P_n^((alpha,beta))(x) or, in terms of hypergeometric functions, as y(x)=C_1_2F_1(-n,n+1+alpha+beta,1+alpha,1/2(x-1)) ...