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Classification of Differential Equations

While differential equations have a basic type of ordinary (ODEs), partial (PDEs), or differential-algebraic (DAEs), they can be further described by attributes such as order, linearity, and degree. The solution method used by DSolve and the nature of the solutions depend heavily on the class of equation being solved.

The order of a differential equation is the order of the highest derivative in the equation.

This is a first-order ODE because its highest derivative is of order 1.

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Here is the general solution to a fourth-order ODE.

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A differential equation is linear if the equation is of the first degree in y and its derivatives, and if their coefficients are functions of the independent variable.

This is a nonlinear second-order ODE that represents the motion of a circular pendulum. It is nonlinear because Sin[y[x]] is not a linear function of y[x]. The Solve::ifun warning message can be ignored here; it appears because Solve uses JacobiAmplitude (the inverse of EllipticF) to find an expression for y[x].

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This plots the solutions.

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It should be noted that sometimes the solutions to fairly simple nonlinear equations are only available in implicit form. In these cases, DSolve returns an unevaluated Solve object.

This nonlinear differential equation only has an implicit solution. The Solve::tdep messages can be ignored; they appear because Solve cannot find an explicit expression for y[x] because non-algebraic functions (ArcTan and Log) are involved.

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When the coefficients of a linear ODE do not depend on x, the ODE is said to have constant coefficients.

This is an ODE with constant coefficients.

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The previous equation is also homogeneous: all terms contain y or derivatives of y and its right-hand side is zero. Adding a function of the independent variable makes the equation inhomogeneous. The general solution to an inhomogeneous equation with constant coefficients is obtained by adding a particular integral to the solution to the corresponding homogeneous equation.

Here,  is added to the right-hand side of the previous equation, making the new equation inhomogeneous. The general solution to this new equation is the sum of the previous solution and a particular integral.

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When the coefficients of an ODE depend on x, the ODE is said to have variable coefficients. Since equations with variable coefficients that are rational functions of x have singularities that are easily classified, there are sophisticated algorithms available for solving them.

The coefficients of this equation are rational functions of x.

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There is a close relationship between functions and differential equations. Starting with a function of almost any type, it is possible to construct a differential equation satisfied by that function. Conversely, any differential equation gives rise to one or more functions, in the form of solutions to that equation. In fact, many special functions from classical analysis have their origins in the solution of differential equations. Mathieu functions are one such class of special functions. Mathieu was interested in studying the vibrations of elliptical membranes. The eigenfunctions for the wave equation that describes this motion are given by products of Mathieu functions.

This linear second-order ODE with rational coefficients has a general solution given by Mathieu functions.

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The presence of ArcCos[t] in the previous solution suggests that the equation can be given a simpler form using trigonometric functions. This is the form in which these equations were introduced by E. Mathieu in 1868.

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This plots the surface for a particular product of solutions to this equation.

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The degree of a differential equation is the highest power of the highest-order derivative in the equation.

This is a first-order ODE of degree 2.

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The higher degree leads to non-uniqueness of the solution.

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The examples in this section have been focused on the classification of ODEs. The classification of PDEs is similar but more involved. PDEs can also be classified by linearity or nonlinearity, order, degree, and constant or variable coefficients. More important is the classification that identifies a PDE as hyperbolic, parabolic, or elliptic. These classifications are discussed in further detail in a later section.