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### 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)`.

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