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### 49.1 Introduction to contrib_ode

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 for linear and non-linear first order ODEs and linear homogeneous second order ODEs. The code is still under development and the calling sequence may change in future releases. Once the code has stabilized it may be moved from the contrib directory and integrated into Maxima.

This package must be loaded with the command `load('contrib_ode)` before use.

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.

The form of the solution differs from `ode2`. As non-linear equations can have multiple solutions, `contrib_ode` returns a list of solutions. Each solution can have a number of forms:

- an explicit solution for the dependent variable,
- an implicit solution for the dependent variable,
- a parametric solution in terms of variable `%t`, or
- a tranformation into another ODE in variable `%u`.

`%c` is used to represent the constant of integration for first order equations. `%k1` and `%k2` are 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.

It is necessary to return a list of solutions, as even first order non-linear ODEs can have multiple solutions. For example:

    (%i1) load('contrib_ode)$

    (%i2) eqn:x*'diff(y,x)^2-(1+x*y)*'diff(y,x)+y=0;
                        dy 2             dy
    (%o2)            x (--)  - (1 + x y) -- + y = 0
                        dx               dx

    (%i3) contrib_ode(eqn,y,x);
                        dy 2             dy
    (%t3)            x (--)  - (1 + x y) -- + y = 0
                        dx               dx

                  first order equation not linear in y'

                                                 x
    (%o3)             [y = log(x) + %c, y = %c %e ]

    (%i4) method;
    (%o4)                        factor

Nonlinear ODEs can have singular solutions without constants of integration, as in the second solution of the following example:

    (%i1) load('contrib_ode)$

    (%i2) eqn:'diff(y,x)^2+x*'diff(y,x)-y=0;
                           dy 2     dy
    (%o2)                 (--)  + x -- - y = 0
                           dx       dx

    (%i3) contrib_ode(eqn,y,x);
                           dy 2     dy
    (%t3)                 (--)  + x -- - y = 0
                           dx       dx

                  first order equation not linear in y'

                                               2
                                     2        x
    (%o3)              [y = %c x + %c , y = - --]
                                              4

    (%i4) method;
    (%o4)                       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.

    (%i1) load('contrib_ode)$

    (%i2) eqn:'diff(y,x)=(x+y)^2;
                              dy          2
    (%o2)                     -- = (x + y)
                              dx

    (%i3) contrib_ode(eqn,y,x);
    (%o3) [[x = %c - atan(sqrt(%t)), y = (- x) - sqrt(%t)],
                         [x = atan(sqrt(%t)) + %c, y = sqrt(%t) - x]]

    (%i4) method;
    (%o4)                       lagrange

The following example (Kamke 1.112) demonstrates an implicit solution.

    (%i1) load('contrib_ode)$

    (%i2) assume(x>0,y>0);
    (%o2)                    [x > 0, y > 0]

    (%i3) eqn:x*'diff(y,x)-x*sqrt(y^2+x^2)-y;
                         dy           2    2
    (%o3)              x -- - x sqrt(y  + x ) - y
                         dx

    (%i4) contrib_ode(eqn,y,x);
                                      y
    (%o4)                  [x - asinh(-) = %c]
                                      x

    (%i5) method;
    (%o5)                          lie

The following Riccati equation is transformed into a linear second order ODE in the variable `%u`. Maxima is unable to solve the new ODE, so it is returned unevaluated.

    (%i1) load('contrib_ode)$

    (%i2) eqn:x^2*'diff(y,x)=a+b*x^n+c*x^2*y^2;
                        2 dy      2  2      n
    (%o2)              x  -- = c x  y  + b x  + a
                          dx

    (%i3) contrib_ode(eqn,y,x);
                   d%u
                   ---                            2
                   dx        2  a       n - 2    d %u
    (%o3)  [[y = - ----, %u c  (-- + b x     ) + ---- c = 0]]
                   %u c          2                 2
                                x                dx

    (%i4) method;
    (%o4)                        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` then `odelin`.

Extensive debugging traces and messages are displayed if the command `put('contrib_ode,true,'verbose)` is executed.

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