ode::series

Series solutions of an ordinary differential equation

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

ode::series(Ly, y(x), x | x = x0, <order>)
ode::series({Ly, <inits>}, y(x), x | x = x0, <order>)

Description

ode::series(Ly, y(x), x = x0) computes the first terms of the series expansions of the solutions of Ly with respect to the variable x around the point x0.

ode::series tries to compute either the Taylor series, the Laurent series or the Puiseux series of the solutions of the differential equation Ly around the point x=x0.

Suppose that Ly is a nonlinear differential equation. If x0 is an ordinary point of Ly then a Taylor series is computed otherwise an expression of type "series" is returned. If initial conditions are given at the point x0 then the answer is expressed in terms of the function y(x) and its derivatives evaluated at the point x0. See Example 1.

Suppose that Ly is a linear differential equation. If x0 is an ordinary point of Ly then a Taylor series is computed, if Ly is furthermore homogeneous and x0 is a regular point then a Puiseux series is computed (containing possible logarithmic terms), otherwise an expression of type "series" is returned. If initial conditions are given at the point x0 then the answer is either expressed in terms of the function y(x) and its derivatives evaluated at the point x0 or it may be expressed in terms of arbitrary constants.

Examples

Example 1

Consider the following nonlinear differential equation:

Ly := x^2*diff(y(x),x)+y(x)-x

We compute the series solutions at the point 0 which is a singular point:

ode::series(Ly, y(x), x=0)

Then we compute the series solutions at the regular point 1:

ode::series(Ly, y(x), x=1)

And we can also put some initial conditions at the point 1:

ode::series({y(1)=1, Ly}, y(x), x=1)

Example 2

Consider the following linear differential equation:

Ly := (2*x+x^3)*diff(y(x),x$2)-diff(y(x),x)-6*x*y(x)

We compute the series solutions at the regular point 1:

ode::series(Ly, y(x), x=1)

The series solutions at the regular singular point 0:

ode::series(Ly, y(x), x=0)

An also the series solutions at the regular singular point infinity:

ode::series(Ly, y(x), x=infinity)

Example 3

Consider the following linear differential equation:

Ly := x^2*diff(y(x),x$2)-x*diff(y(x),x)+(1-x)*y(x)

We compute the series solutions at the regular singular point 0:

ode::series(Ly, y(x), x)

And at the same point we look for solutions satisfying the initial condition y(0) = 1 and y(0) = 0:

ode::series({y(0)=1, Ly}, y(x), x)

ode::series({y(0)=0, Ly}, y(x), x)

Parameters

Ly

An ordinary differential equation.

y(x)

The dependent function of Ly.

x

The independent variable of Ly.

x0

The expansion point: an arithmetical expression. If not specified, the default expansion point 0 is used .

inits

The initial or boundary conditions: a sequence of equations.

order

The number of terms to be computed: a nonnegative integer. The default order is given by the environment variable ORDER (default value 6).

Return Values

Either a list, maybe empty, of objects of type Series::Puiseux or an expression of type "series".

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