# Documentation

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# Ordinary Differential Equations

Solve ODEs analytically, test solutions

 `ode::exponentialSolutions` Exponential solutions of a homogeneous linear ordinary differential equation `ode::polynomialSolutions` Polynomial solutions of a homogeneous linear ordinary differential equation `ode::rationalSolutions` Rational solutions of a homogeneous linear ordinary differential equation `ode::series` Series solutions of an ordinary differential equation `ode::solve` Solving ordinary differential equations
 `ode` Domain of ordinary differential equations `ode::companionSystem` Companion matrix of a linear homogeneous ordinary differential equation `ode::cyclicVector` Transforms a linear differential system to an equivalent linear differential system with a companion matrix. `ode::dAlembert` D'Alembert reduction of a linear homogeneous ordinary differential equation `ode::evalOde` Applies an expression at a linear ordinary differential equation `ode::exponents` Exponents of a linear ordinary differential equation `ode::getOrder` Order of an ordinary differential equation `ode::indicialEquation` Indicial equation of a linear ordinary differential equation `ode::isFuchsian` Tests if a homogeneous linear ordinary differential equation is of Fuchsian type `ode::isLODE` Test for a linear ordinary differential equation `ode::mkODE` Builds a linear homogeneous ordinary differential equation from a list of coefficient functions `ode::normalize` Normalized form of a linear ordinary differential equation `ode::ratSys` Rational solutions of a first order homogeneous linear differential system `ode::scalarEquation` Transforms a linear differential system to an equivalent scalar linear differential equation `ode::symmetricPower` Symmetric power of a homogeneous linear ordinary differential equation `ode::unimodular` Unimodular transformation of a linear ordinary differential equation `ode::vectorize` Coefficients of a homogeneous linear ODE `ode::wronskian` Wronskian of functions or of a linear homogeneous ordinary differential equation

## Examples and How To

Choose a Solver

The general solvers (`solve` for symbolic solutions and `numeric::solve` for numeric approximations) handle a wide variety of equations, inequalities, and systems.

Solve Ordinary Differential Equations and Systems

An ordinary differential equation (ODE) contains derivatives of dependent variables with respect to the only independent variable.

Test Results

Suppose you want to verify the solutions of this polynomial equation:

## Concepts

If Results Look Too Complicated

By default, the MuPAD® solvers return all possible solutions regardless of their length.

If Results Differ from Expected

Symbolic solutions can be returned in different, but mathematically equivalent forms.