(maxima.info)Functions and Variables for contrib_ode

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49.2 Functions and Variables for contrib_ode
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 -- 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);
                 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)                   --- + (b + a x) 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 (b + a x)
          (%o3) [y = bessel_y(-, --------------) %k2 sqrt(a x + b)
                              3       3 a
                                                    3/2
                                      1  2 (b + a x)
                           + 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>)'.

 -- Function: bessel_simplify (<expr>)
     Simplifies expressions containing Bessel functions bessel_j,
     bessel_y, bessel_i, bessel_k, hankel_1, hankel_2, strauve_h and
     strauve_l.  Recurrence relations (given in Abramowitz and Stegun,
     Handbook of Mathematical Functions, Section 9.1.27) are used to
     replace functions of highest order n by functions of order n-1 and
     n-2.

     This process repeated until all the orders differ by less than 2.

          (%i1) load('contrib_ode)$
          (%i2) bessel_simplify(4*bessel_j(n,x^2)*(x^2-n^2/x^2)
            +x*((bessel_j(n-2,x^2)-bessel_j(n,x^2))*x
            -(bessel_j(n,x^2)-bessel_j(n+2,x^2))*x)
            -2*bessel_j(n+1,x^2)+2*bessel_j(n-1,x^2));
          (%o2)                           0
          (%i3) bessel_simplify(-2*bessel_j(1,z)*z^3-10*bessel_j(2,z)*z^2
           +15*%pi*bessel_j(1,z)*struve_h(3,z)*z-15*%pi*struve_h(1,z)*bessel_j(3,z)*z
           -15*%pi*bessel_j(0,z)*struve_h(2,z)*z+15*%pi*struve_h(0,z)*bessel_j(2,z)*z
           -30*%pi*bessel_j(1,z)*struve_h(2,z)+30*%pi*struve_h(1,z)*bessel_j(2,z));
          (%o3)                           0

 -- Function: expintegral_e_simplify (<expr>)
     Simplify expressions containing exponential integral expintegral_e
     using the recurrence (A&S 5.1.14).

     expintegral_e(n+1,z) = (1/n) * (exp(-z)-z*expintegral_e(n,z)) n =
     1,2,3 ....