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The ordinary Onsager equation is the sixth-order ordinary differential equation (d^3)/(dx^3)[e^x(d^2)/(dx^2)(e^x(dy)/(dx))]=f(x) (Vicelli 1983; Zwillinger 1997, p. 128), ...

y^('')-mu(1-1/3y^('2))y^'+y=0, where mu>0. Differentiating and setting y=y^' gives the van der Pol equation. The equation y^('')-mu(1-y^('2))y^'+y=0 with the 1/3 replaced by ...

The differential equation where alpha+alpha^'+beta+beta^'+gamma+gamma^'=1, first obtained in the form by Papperitz (1885; Barnes 1908). Solutions are Riemann P-series ...

The equation of motion for a membrane shaped as a right isosceles triangle of length c on a side and with the sides oriented along the positive x and y axes is given by where ...

The Bessel differential equation is the linear second-order ordinary differential equation given by x^2(d^2y)/(dx^2)+x(dy)/(dx)+(x^2-n^2)y=0. (1) Equivalently, dividing ...

Cauchy's functional equation is the equation f(x+y)=f(x)+f(y). It was proved by Cauchy in 1821 that the only continuous solutions of this functional equation from R into R ...

A universal differential equation (UDE) is a nontrivial differential-algebraic equation with the property that its solutions approximate to arbitrary accuracy any continuous ...

The Diophantine equation sum_(j=1)^(m-1)j^n=m^n. Erdős conjectured that there is no solution to this equation other than the trivial solution 1^1+2^1=3^1, although this ...

z(1-z)(d^2y)/(dz^2)+[c-(a+b+1)z](dy)/(dz)-aby=0. It has regular singular points at 0, 1, and infty. Every second-order ordinary differential equation with at most three ...

The ordinary differential equation (1) (Byerly 1959, p. 255). The solution is denoted E_m^p(x) and is known as an ellipsoidal harmonic of the first kind, or Lamé function. ...