Series solutions of an ordinary differential equation
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ode::series(Ly
, y(x
),x  x = x0
, <order
>) ode::series({Ly, <inits>}
, y(x
),x  x = x0
, <order
>)
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.
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)
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)
Consider the following linear differential equation:
Ly := x^2*diff(y(x),x$2)x*diff(y(x),x)+(1x)*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)

An ordinary differential equation. 

The dependent function of 

The independent variable of 

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

The initial or boundary conditions: a sequence of equations. 

The number of terms to be computed: a nonnegative integer. The
default order is given by the environment variable 
Either a list
, maybe empty, of objects of type Series::Puiseux
or an
expression of type "series"
.