Solution of Differential Equations
		    by use of Laplace Transforms


    DIFSOLVE finds the solution of differential equations or systems of
them by use of Laplace transforms.  Currently it is limited to constant-
coefficient linear differential equations.  If initial conditions on the
independent variable are given, they are applied at zero and are imposed
on the given equation.  The functional dependencies of
the variables in the given equations must be declared before DIFSOLVE is
called.  Only one independent variable may appear in the supplied equa-
tions.  The dependent variables are not specified as arguments because
these variables are obtained from the LISTOFVARS function.  Note that
any undetermined coefficients must be declared to be constant, e.g.
DECLARE(a,CONSTANT).

    The input parameters are not checked for correctness.  However, the
given equations should be checked to insure that they are linear.  Also
the coefficients should be checked to insure that they are constant.
Additional checks that should be done are that the independent variable
and all its derivatives (1) are raised only to the first power and (2)
never appear as cross-products.

    For an nth-order differential equation, there must be n initial con-
ditions if a unique solution is to be obtained.  When non-zero initial
values are given, the final function determining the solution contains
additional terms in the numerator.


			Algorithm
			=========

    First the routine declares the functional relationships of the de-
pendent variables explicitly.  Then the initial conditions are applied.
The Laplace transform is taken, and any boundary conditions are imposed
on the resulting set of simultaneous linear equations.  Then the inde-
pendent variable is solved for.  Taking the inverse transform gives the
final solution.  Note that Macsyma will be able to find the Laplace
transform of most of the differential equations of interest.  However
the inverse transform is less likely to be known.  Also the solution is
dependent on the success of the linear equation solver in Macsyma.

    The DIFSOLVE function takes one arguments:
	1)  a differential equation, or if a system of equations
	    is to be solved, then a list of equations

		Future planned enhancements:

1.  If the inverse Laplace transform fails, use the technique of partial
fractions to break the fraction into several fractions with denominators
of quadratic degree or less.

2.  Entend to include boundary conditions that would be imposed on the
transformed equations, which are a set of simultaneous linear equations.

3.  Extend to include initial conditions not given at x=0.  The equation
would be solved without the initial conditions.  The given initial con-
ditions would then be used to solve for the undetermined coefficients.

    I haven't written the Macsyma program yet, but I would like
to receive comments and suggestions.  Mail should be sent to
MSC@MIT-MC.