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常微分方程式

ODE の解析的求解、解のテスト

メモ

MuPAD® Notebook は将来のリリースでは削除される予定です。代わりに MATLAB® ライブ スクリプトを使用してください。

MuPAD Notebook ファイルを MATLAB ライブ スクリプト ファイルに変換するには、convertMuPADNotebook を参照してください。MATLAB ライブ スクリプトは、多少の違いはありますが、MuPAD 機能の大半をサポートします。詳細は、MuPAD Notebook を MATLAB ライブ スクリプトに変換を参照してください。

MuPAD 関数

ode::exponentialSolutionsExponential solutions of a homogeneous linear ordinary differential equation
ode::polynomialSolutionsPolynomial solutions of a homogeneous linear ordinary differential equation
ode::rationalSolutionsRational solutions of a homogeneous linear ordinary differential equation
ode::seriesSeries solutions of an ordinary differential equation
ode::solveSolving ordinary differential equations
odeDomain of ordinary differential equations
ode::companionSystemCompanion matrix of a linear homogeneous ordinary differential equation
ode::cyclicVectorTransforms a linear differential system to an equivalent linear differential system with a companion matrix.
ode::dAlembertD'Alembert reduction of a linear homogeneous ordinary differential equation
ode::evalOdeApplies an expression at a linear ordinary differential equation
ode::exponentsExponents of a linear ordinary differential equation
ode::getOrderOrder of an ordinary differential equation
ode::indicialEquationIndicial equation of a linear ordinary differential equation
ode::isFuchsianTests if a homogeneous linear ordinary differential equation is of Fuchsian type
ode::isLODETest for a linear ordinary differential equation
ode::mkODEBuilds a linear homogeneous ordinary differential equation from a list of coefficient functions
ode::normalizeNormalized form of a linear ordinary differential equation
ode::ratSysRational solutions of a first order homogeneous linear differential system
ode::scalarEquationTransforms a linear differential system to an equivalent scalar linear differential equation
ode::symmetricPowerSymmetric power of a homogeneous linear ordinary differential equation
ode::unimodularUnimodular transformation of a linear ordinary differential equation
ode::vectorizeCoefficients of a homogeneous linear ODE
ode::wronskianWronskian of functions or of a linear homogeneous ordinary differential equation

例および操作のヒント

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. When you use the general solver, MuPAD identifies the equation or the system as one of the types listed in the table that follows. Then the system calls the appropriate solver for that type. If you know the type of the equation or system you want to solve, directly calling the special solver is more efficient. When you call special solvers, MuPAD skips trying other solvers. Direct calls to the special solvers can help you to:

Solve Ordinary Differential Equations and Systems

An ordinary differential equation (ODE) contains derivatives of dependent variables with respect to the only independent variable. If y is a dependent variable and x is an independent variable, the solution of an ODE is an expression y(x). The order of the derivative of a dependent variable defines the order of an ODE.

Test Results

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

概念

If Results Look Too Complicated

By default, the MuPAD solvers return all possible solutions regardless of their length. Also, by default the solvers assume the solutions are complex numbers. To limit the number of the solutions to some specific ones, the solvers provide a number of options. For information about the options accepted by a particular solver, see the page for that solver. For example, for the list of options provided by the general solver, see the solve help page.

If Results Differ from Expected

Symbolic solutions can be returned in different, but mathematically equivalent forms. MuPAD continuously improves its functionality, including solvers and simplifiers. These improvements can cause different releases of MuPAD to return different forms of the same symbolic expressions. For example, when you solve the equation