38.2 Functions and Variables for contrib_ode

Function: contrib_ode (eqn, y, x) ¶

Returns a list of solutions of the ODE eqn with independent variable x and dependent variable y.

Function: odelin (eqn, y, x) ¶

odelin solves linear homogeneous ODEs of first and second order with independent variable x and dependent variable y. It returns a fundamental solution set of the ODE.

For second order ODEs, odelin uses a method, due to Bronstein and Lafaille, that searches for solutions in terms of given special functions.

(%i1) load("contrib_ode");

(%i2) odelin(x*(x+1)*'diff(y,x,2)+(x+5)*'diff(y,x,1)+(-4)*y,y,x);
...trying factor method
...solving 7 equations in 4 variables
...trying the Bessel solver
...solving 1 equations in 2 variables
...trying the F01 solver
...solving 1 equations in 3 variables
...trying the spherodial wave solver
...solving 1 equations in 4 variables
...trying the square root Bessel solver
...solving 1 equations in 2 variables
...trying the 2F1 solver
...solving 9 equations in 5 variables
       gauss_a(- 6, - 2, - 3, - x)  gauss_b(- 6, - 2, - 3, - x)
(%o2) {---------------------------, ---------------------------}
                    4                            4
                   x                            x

Function: ode_check (eqn, soln) ¶

Returns the value of ODE eqn after substituting a possible solution soln. The value is equivalent to zero if soln is a solution of eqn.

(%i1) load("contrib_ode")$
(%i2) eqn:'diff(y,x,2)+(a*x+b)*y;

                         2
                        d y
(%o2)                   --- + (a x + b) y
                          2
                        dx
(%i3) ans:[y = bessel_y(1/3,2*(a*x+b)^(3/2)/(3*a))*%k2*sqrt(a*x+b)
         +bessel_j(1/3,2*(a*x+b)^(3/2)/(3*a))*%k1*sqrt(a*x+b)];

                                  3/2
                    1  2 (a x + b)
(%o3) [y = bessel_y(-, --------------) %k2 sqrt(a x + b)
                    3       3 a
                                          3/2
                            1  2 (a x + b)
                 + bessel_j(-, --------------) %k1 sqrt(a x + b)]
                            3       3 a
(%i4) ode_check(eqn,ans[1]);
(%o4)                           0

System variable: method ¶

The variable method is set to the successful solution method.

Variable: %c ¶

%c is the integration constant for first order ODEs.

Variable: %k1 ¶

%k1 is the first integration constant for second order ODEs.

Variable: %k2 ¶

%k2 is the second integration constant for second order ODEs.

Function: gauss_a (a, b, c, x) ¶

gauss_a(a,b,c,x) and gauss_b(a,b,c,x) are 2F1 geometric functions. They represent any two independent solutions of the hypergeometric differential equation x(1-x) diff(y,x,2) + [c-(a+b+1)x diff(y,x) - aby = 0 (A&S 15.5.1).

The only use of these functions is in solutions of ODEs returned by odelin and contrib_ode. The definition and use of these functions may change in future releases of Maxima.

See also gauss_b, dgauss_a and gauss_b.

Function: gauss_b (a, b, c, x) ¶

See gauss_a.

Function: dgauss_a (a, b, c, x) ¶

The derivative with respect to x of gauss_a(a, b, c, x).

Function: dgauss_b (a, b, c, x) ¶

The derivative with respect to x of gauss_b(a, b, c, x).

Function: kummer_m (a, b, x) ¶

Kummer’s M function, as defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 13.1.2.

The only use of this function is in solutions of ODEs returned by odelin and contrib_ode. The definition and use of this function may change in future releases of Maxima.

See also kummer_u, dkummer_m and dkummer_u.

Function: kummer_u (a, b, x) ¶

Kummer’s U function, as defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 13.1.3.

See kummer_m.

Function: dkummer_m (a, b, x) ¶

The derivative with respect to x of kummer_m(a, b, x).

Function: dkummer_u (a, b, x) ¶

The derivative with respect to x of kummer_u(a, b, x).