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-*- 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.
A solution can have a number of forms:
o an explicit solution for the dependent variable,
o an implicit solution for the dependent variable,
o a parametric solution in terms of variable %t, or
o a tranformation into another ODE in variable %u.
%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
The following Riccati equation is transformed into a linear
second order ODE in the variable %u. maxima is unable to
solve the new ODE.
(%i7) eqn:'diff(y,x)+1-x=(x+y)*y;
dy
(%o7) -- - x + 1 = y (y + x)
dx
(%i8) contrib_ode(eqn,y,x);
d%u
--- 2
dx d%u D %u
(%o8) [[y = - ---, - --- x + %u (x - 1) + ---- = 0]]
%u dx 2
dx
(%i9) method;
(%o9) riccati
For first order ODEs contrib_ode calls ode2. It then tries the
following methods: factorization, Clairault, Lagrange, Riccati,
Abel and Lie symmetry methods. The Lie method is not attempted
on Abel equations if the Abel method fails, but it is tried
if the Riccati method returns an unsolved second order ODE.
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.
* Improve the detection of Abel equations of second kind. The exisiting
pattern matching is weak.
* 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
[4] F Schwarz, Symmetry Analysis of Abel's Equation, Studies in
Applied Mathematics, 100:269-294 (1998)
[5] F Schwarz, Algorithmic Solution of Abel's Equation,
Computing 61, 39-49 (1998)
[6] E. S. Cheb-Terrab, A. D. Roche, Symmetries and First Order
ODE Patterns, Computer Physics Communications 113 (1998), p 239.
(http://lie.uwaterloo.ca/papers/ode_vii.pdf)
[7] E. S. Cheb-Terrab, T. Koloknikov, First Order ODEs,
Symmetries and Linear Transformations, European Journal of
Applied Mathematics, Vol. 14, No. 2, pp. 231-246 (2003).
(http://arxiv.org/abs/math-ph/0007023)
(http://lie.uwaterloo.ca/papers/ode_iv.pdf)
[8] G W Bluman, S C Anco, Symmetry and Integration Methods for
Differential Equations, Springer, (2002)
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