Differential Equations Differential Equations

30.1: Definitions for Differential Equations

Function: DESOLVE ([eq1,...,eqn],[var1,...,varn])

where the eq's are differential equations in the dependent variables var1,...,varn. The functional relationships must be explicitly indicated in both the equations and the variables. For example

(C1) 'DIFF(F,X,2)=SIN(X)+'DIFF(G,X);
(C2) 'DIFF(F,X)+X^2-F=2*'DIFF(G,X,2);
is NOT the proper format.  The correct way is:
(C3) 'DIFF(F(X),X,2)=SIN(X)+'DIFF(G(X),X);
(C4) 'DIFF(F(X),X)+X^2-F(X)=2*'DIFF(G(X),X,2);
The call is then DESOLVE([D3,D4],[F(X),G(X)]);
If initial conditions at 0 are known, they should be supplied before
calling DESOLVE by using ATVALUE.
(C11) 'DIFF(F(X),X)='DIFF(G(X),X)+SIN(X);
                        d         d
(D11)                   -- F(X) = -- G(X) + SIN(X)
                        dX        dX
(C12) 'DIFF(G(X),X,2)='DIFF(F(X),X)-COS(X);
                         2
                        d          d
(D12)                   --- G(X) = -- F(X) - COS(X)
                          2        dX
                        dX
(C13) ATVALUE('DIFF(G(X),X),X=0,A);
(D13)                                A
(C14) ATVALUE(F(X),X=0,1);
(D14)                                1
(C15) DESOLVE([D11,D12],[F(X),G(X)]);
                X                            X
(D16) [F(X)=A %E  - A+1, G(X) = COS(X) + A %E  - A + G(0) - 1]
/* VERIFICATION */
(C17) [D11,D12],D16,DIFF;
                  X       X      X                X
(D17)        [A %E  = A %E , A %E  - COS(X) = A %E  - COS(X)]

If DESOLVE cannot obtain a solution, it returns "FALSE".

Function: IC1 (exp,var,var)

In order to solve initial value problems (IVPs) and boundary value problems (BVPs), the routine IC1 is available in the ODE2 package for first order equations, and IC2 and BC2 for second order IVPs and BVPs, respectively. Do LOAD(ODE2) to access these. They are used as in the following examples:

(C3) IC1(D2,X=%PI,Y=0);
                         COS(X) + 1
(D3)               Y = - ----------
                              3
                             X
(C4) 'DIFF(Y,X,2) + Y*'DIFF(Y,X)^3 = 0;
                       2
                      d Y      dY 3
(D4)                  --- + Y (--)  = 0
                        2      dX
                      dX
(C5) ODE2(%,Y,X);
                 3
                Y  - 6 %K1 Y - 6 X
(D7)            ------------------ = %K2
                        3
(C8) RATSIMP(IC2(D7,X=0,Y=0,'DIFF(Y,X)=2));
                     3
                  2 Y  - 3 Y + 6 X
(D9)            - ---------------- = 0
                         3
(C10) BC2(D7,X=0,Y=1,X=1,Y=3);
                 3
                Y  - 10 Y - 6 X
(D11)           --------------- = - 3
                       3

Function: ODE (equation,y,x)

a pot-pourri of Ordinary Differential solvers combined in such a way as to attempt more and more difficult methods as each fails. For example, the first attempt is with ODE2, so therefore, a user using ODE can assume he has all the capabilities of ODE2 at the very beginning and if he has been using ODE2 in programs they will still run if he substitutes ODE (the returned values, and calling sequence are identical). In addition, ODE has a number of user features which can assist an experienced ODE solver if the basic system cannot handle the equation. The equation is of the same form as required for ODE2 (which see) and the y and x are dependent and independent variables, as with ODE2. For more details, do PRINTFILE(ODE,USAGE,SHARE); .

Function: ODE2 (exp,dvar,ivar)

takes three arguments: an ODE of first or second order (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 either an explicit or implicit solution for the dependent variable. %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 ODE2 cannot obtain a solution for whatever reason, it returns FALSE, after perhaps printing out an error message. The methods implemented for first order equations in the order in which they are tested are: linear, separable, exact - perhaps requiring an integrating factor, homogeneous, Bernoulli's equation, and a generalized homogeneous method. For second order: constant coefficient, exact, linear homogeneous with non-constant coefficients which can be transformed to constant coefficient, the Euler or equidimensional equation, the method of variation of parameters, and equations which are free of either the independent or of the dependent variable so that they can be reduced to two first order linear equations to be solved sequentially. In the course of solving ODEs, several variables are set purely for informational purposes: METHOD denotes the method of solution used e.g. LINEAR, INTFACTOR denotes any integrating factor used, ODEINDEX denotes the index for Bernoulli's method or for the generalized homogeneous method, and YP denotes the particular solution for the variation of parameters technique.

Differential Equations

Differential Equations