MACSYMA's ordinary differential equation (ODE) solver ODE2 may be used for symbolically solving elementary ODEs of first and second order. One calls the ODE2 routine as follows: (C1) X^2*'DIFF(Y,X) + 3*X*Y = SIN(X)/X; 2 dY SIN(X) (D1) X -- + 3 X Y = ------ dX X (C2) ODE2(%,Y,X); %C - COS(X) (D2) Y = ----------- 3 X We see from this example how ODE2 is used. Namely, it 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. In order to solve initial value problems (IVPs) and boundary value problems (BVPs), the routine IC1 is available for first order equations, and IC2 and BC2 for second order IVPs and BVPs, respectively. 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 In order to see more clearly which methods have been implemented, a demonstration file is available. To run it, you may do DEMO(ODE2,DEMO,DSK,SHARE); .