Contributing some first order ode routines

I have been converting Alan Head's Lie53 symmetry code to maxima.
Can we integrate this code with yours?

"Billinghurst, David (CALCRTS)" wrote:

> Over my summer break I have managed to work on my first order ode
> routines.  I have attached a brief description below.  I'd like to
> create a share/contrib/diffequations directory and put them there
> to mature for a while.  Any objections.
>
> ========================== contrib_ode.usg ========================
> -*- mode: Text;  -*-
>
> MAXIMA's ordinary differential equation (ODE) solver ODE2 solves
> elementary linear ODEs of first and second order.  The function
> contrib_ode extends ODE2 with additional methods.  The code may be
> integrated into MAXIMA at some stage.
>
> The calling convention for contrib_ode is identical to ODE2.  It takes
> three arguments: an ODE (only the left hand side need be given if the
> right hand side is 0), the dependent variable, and the independent
> variable.  When successful, it returns a list of solutions.  Each
> solution can be: an explicit solution for the dependent variable; an
> implicit solution for the dependent variable; or a parametric solution
> in terms of variable %t.  %C is used to represent the constant in the
> case of first order equations, and %K1 and %K2 the constants for
> second order equations.  If contrib_ode cannot obtain a solution for
> whatever reason, it returns FALSE, after perhaps printing out an error
> message.
>
> (C1) eqn:x*'diff(y,x)^2-(1+x*y)*'diff(y,x)+y=0;
>
>                            dy 2             dy
> (D1)                    x (--)  - (x y + 1) -- + y = 0
>                            dx               dx
>
> (C2) contrib_ode(eqn,y,x);
>
>                                                     x
> (D2)                     [y = LOG(x) + %C, y = %C %E ]
>
> (C3) method;
>
> (D3)                                FACTOR
>
> Nonlinear odes can have singular solutions without constants of
> integration.
>
> (C4) eqn:'diff(y,x)^2+x*'diff(y,x)-y=0;
>
>                               dy 2     dy
> (D4)                         (--)  + x -- - y = 0
>                               dx       dx
>
> (C5) contrib_ode(eqn,y,x);
>
>                                                   2
>                                         2        x
> (D5)                      [y = %C x + %C , y = - --]
>                                                  4
> (C6) method;
>
> (D6)                               clairault
>
> The following ode has two parametric solutions in terms of the dummy
> variable %t.  In this case the parametric solutions can be manipulated
> to give explicit solutions.
>
> (C7) eqn:'diff(y,x)=(x+y)^2;
>
>                                  dy          2
> (D7)                             -- = (y + x)
>                                  dx
>
> (C8) contrib_ode(eqn,y,x);
>
> (D8) [[x = %C - ATAN(SQRT(%T)), y = - x - SQRT(%T)],
>       [x = ATAN(SQRT(%T)) + %C, y = SQRT(%T) - x]]
>
> (C9) method;
>
> (D9)                              lagrange
>
> For first order ODEs contrib_ode calls ode2.  It then tries the
> following methods: factorization, Clairault, Lagrange, Riccati and Lie
> symmetry methods.
>
> For second order ODEs contrib_ode calls ode2.  No other methods are
> implemented yet.
>
> Extensive debugging traces and messages are displayed if the command
> put('contrib_ode,true,'verbose) is executed.
>
> TO DO
> =====
>
> These routines are work in progress.  I still need to:
>
> * Extend the FACTOR method ode1_factor to work for multiple roots.
>
> * Extend the FACTOR method ode1_factor to attempt to solve higher
>   order factors.  At present it only attemps to solve linear factors.
>
> * Fix the LAGRANGE routine ode1_lagrange to prefer real roots over
>   complex roots.
>
> * Add additional methods for Riccati equations.
>
> * Work out how to return partial solutions, such as those obtained for
>   Riccati equations.
>
> * Add routines for Abel equations.
>
> * Work on the Lie symmetry group routine ode1_lie.  There are quite a
>   few problems with it: some parts are unimplemented; some test cases
>   seem to run forever; other test cases crash; yet others return very
>   complex "solutions".  I wonder if it really ready for release yet.
>
> * Add more test cases.
>
> TEST CASES
> ==========
>
> The routines have been tested on a few hundred test cases from Murphy,
> Kamke and Zwillinger.  These are included in the tests subdirectory.
>
> * The Clairault routine ode1_clairault finds all known solutions,
>   including singular soultions.  (This statement is asking for
>   trouble).
>
> * The other routines often return a single solution when multiple
>   solutions exist.
>
> * Some of the "solutions" from ode1_lie are overly complex and
>   impossible to check.
>
> * There are some crashes.
>
> References
> ==========
>
> [1] E Kamke, Differentialgleichungen Losungsmethoden und Losungen, Vol 1,
>     Geest & Portig, Leipzig, 1961
>
> [2] G M Murphy, Ordinary Differential Equations and Their Solutions,
>     Van Nostrand, New York, 1960
>
> [3] D Zwillinger, Handbook of Differential Equations, 3rd edition,
>     Academic Press, 1998
>
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