An Euler equation is an ODE of the form The following is an example of an Euler equation. The Legendre linear equation is a generalization of the Euler equation. It has the ...

There are four major areas in the study of ordinary differential equations that are of interest in pure and applied science. Of these four areas, the study of exact solutions ...

Mathematica uses a large number of original algorithms to provide automatic systemwide support for inequalities and inequality constraints. Whereas equations can often be ...

A linear ODE with constant coefficients can be easily solved once the roots of the auxiliary equation (or characteristic equation) are known. Some examples of this type ...

A Frobenius equation is an equation of the form where a_1, …, a_n are positive integers, m is an integer, and the coordinates x_1, …, x_n of solutions are required to be ...

The Lotka–Volterra system arises in mathematical biology and models the growth of animal species. Consider two species where Y_1(T) denotes the number of predators and Y_2(T) ...

The solutions to many second-order ODEs can be expressed in terms of special functions. Solutions to certain higher-order ODEs can also be expressed using AiryAi, BesselJ, ...

LQOutputRegulatorGains[ss, {q, r}] gives the optimal state feedback gain matrix for the StateSpaceModel object ss and the quadratic cost function with output and control ...

LQRegulatorGains[ss, {q, r}] gives the optimal state feedback gain matrix for the StateSpaceModel object ss and the quadratic cost function with state and control weighting ...

DSolve can find solutions for most of the standard linear second-order ODEs that occur in applied mathematics. Here is the solution for Airy's equation. Here is a plot that ...