ode2singular.wxm

/* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/
/* [ Created with wxMaxima version 23.05.1 ] */
/* [wxMaxima: comment start ]
The function ode2singular tries to find the general solution y(x) to the equation p2(x)*y''(x)+p1(x)*y'(x)+p0(x)*y(x)=0.
Load the file ode2singular.mac. We also load contrib_ode in order to use the function ode_check.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
kill(all);
load('ode2singular);
load('contrib_ode);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
By default, ode2singular determines fundamental systems of solutions to homogeneous equations accurately by distinguishing quite a few cases. This leads sometimes to a long computation time and to many questions asked to the user. This can be deactivated by putting "testmode: true" which we do in the following. Notice however that with "testmode: true" some inhomogeneous equations cannot be solved which can be solved with the default setting.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
testmode: true;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: section start ]
Constant coefficients and power functions
   [wxMaxima: section end   ] */


/* [wxMaxima: input   start ] */
p2: 1$
p1: 0$
p0: 4$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=x$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
nonzero;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
p2: 1$
p1: 0$
p0: -2/x^2$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=x$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: section start ]
Parabolic cylinder functions, Hermite equation
   [wxMaxima: section end   ] */


/* [wxMaxima: input   start ] */
kill(a)$
p2: 1$
p1: 0$
p0: -(x-8)^(2*a-2)+(x-8)^(a-2)$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=0$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #2
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #7
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #3
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #5
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #8
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #6
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #4
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[2]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
kill(phi)$
p2: 1$
p1: 0$
p0: -125*(phi-2)$
eq: p2*'diff(y,phi,2)+p1*'diff(y,phi)+p0*y=0$
ans: ode2singular(eq,y,phi);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[2]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[3]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
kill(a)$
p2: 1$
p1: 0$
p0: -(x-5)^a$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=0$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #2
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #3
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[2]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[3]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
For the Hermite equation ode2singular gives solutions in terms of Hermite polynomials:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
lambda: 4$
p2: 1$
p1: -2*x$
p0: 2*lambda$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=0$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #2
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
yes;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
yes;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[2]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: section start ]
Mathieu equation and spheroidal wave equation
   [wxMaxima: section end   ] */


/* [wxMaxima: comment start ]
ode2singular finds solutions which are spheroidal wave functions and also checks whether they are Mathieu functions.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
a: 3$
q: 5$
p2: 1-x^2$
p1: -x$
p0: a+q*(2-4*x^2)$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=0$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #2
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
neg;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #3
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
neg;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
neg;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #4
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
b: 2$
c: 6$
q: 8$
p2: 1-x^2$
p1: -2*(b+1)*x$
p0: c-4*q*x^2$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=0$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #2
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #3
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: section start ]
Bessel equation and Kummer equation
   [wxMaxima: section end   ] */


/* [wxMaxima: comment start ]
ode2singular finds solutions which are the composition of Bessel or Kummer functions with a power of a Mobius transformation.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
p2: 4*(x-1)^2*(x-3)^6$
p1: 0$
p0: 4*(x-1)^4-15*(x-3)^4$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=0$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[2]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[3]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
The Bessel equation:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
a: 1$
p2: x^2$
p1: x$
p0: x^2-a^2$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=0$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[2]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[3]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
The Kummer equation: ode2singular gives solutions in terms of Laguerre functions if a is a negative integer
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
a: -1$
b: 1$
p2: x$
p1: b-x$
p0: -a$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=0$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[2]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
A Bessel function applied to a fractional power:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
p2: 36*x^2$
p1: 0$
p0: 36*x^(2/3)+5$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=0$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[2]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[3]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: section start ]
Hypergeometric equation
   [wxMaxima: section end   ] */


/* [wxMaxima: comment start ]
ode2singular gives the solutions in terms of Legendre functions if the equation can be reduced to a Legendre equation.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
mu: 1$
nu: 2$
p2: 1-x^2$
p1: -2*x$
p0: nu*(nu+1)-mu^2/(1-x^2)$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=0$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #2
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
neg;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
a: 1$
b: 0$
c: -3$
p2: x*(1-x)$
p1: c-(a+b+1)*x$
p0: -a*b$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=0$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #2
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #3
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
neg;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
neg;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #4
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
kill(m)$
declare(m, integer)$
p2: 1$
p1: (m+x)/(x*(1+x))$
p0: -(m+1)*m*(m-1)/(2*(2*m-1)*x*(1+x)^2)$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=0$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #2
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #7
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #3
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #5
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
neg;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #6
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #4
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
no;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
An example of an equation with 4 regular singular points:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
p2: 4*(x+1)^2*x^2*(x-1)^2$
p1: 0$
p0: x^2+1$
eq: p2*'diff(y,x,2)+p1*'diff(y,x)+p0*y=0$
ans: ode2singular(eq,y,x);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #2
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
neg;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
neg;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: section start ]
Change of variable
   [wxMaxima: section end   ] */


/* [wxMaxima: comment start ]
ode2singular finds the solutions to the Mathieu equation if the user enters a suitable change of variable and its inverse:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
kill(a)$
kill(q)$
p2: 1$
p1: 0$
p0: a-2*q*cos(2*t)$
eq: p2*'diff(z,t,2)+p1*'diff(z,t)+p0*z=0$
ans: ode2singular(eq,z,t);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #2
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
neg;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
phi: cos(t)$
phi_inv: acos(t)$
ans: ode2singular_transform(eq,z,t,phi,phi_inv);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #2
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
neg;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
ode_check(eq,ans[1]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
The following equation is solved by spherical harmonic functions. ode2singular finds the solutions if the user enters a suitable change of variable and its inverse.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
kill(ell)$
kill(m)$
declare(m,integer)$
declare(ell,integer)$
p2: sin(theta)^2$
p1: cos(theta)*sin(theta)$
p0: ell*(ell+1)*sin(theta)^2-m^2$
eq: p2*'diff(y,theta,2)+p1*'diff(y,theta)+p0*y=0$
ans: ode2singular(eq,y,theta);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #2
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
yes;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #3
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
nonzero;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: input   start ] */
testmode: false;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
phi: cos(theta)$
phi_inv: acos(theta)$
ans: ode2singular_transform(eq,y,theta,phi,phi_inv);
/* [wxMaxima: input   end   ] */
/* [wxMaxima: question  start ] */
Question #11
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
yes;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #9
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
yes;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #2
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #7
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #10
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #3
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
neg;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #5
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #8
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
yes;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #14
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #1
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
pos;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #6
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
neg;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #4
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
nonzero;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #12
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
yes;
/* [wxMaxima: answer  end   ] */
/* [wxMaxima: question  start ] */
Question #13
/* [wxMaxima: question  end   ] */
/* [wxMaxima: answer  start ] */
yes;
/* [wxMaxima: answer  end   ] */


/* [wxMaxima: comment start ]
Let f be a solution to a linear homogeneous ode of order 2. We write f = h * g where g satisfies an ode of the form g'' + q0 * g = 0. The function transform computes the functions h and q0 after a change of variables replacing x by phi(x).
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
values;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
kill(k)$
kill(x0)$
kill(alpha)$
h: exp(-3*x)$
q0: -x$
phi: alpha*(x-x0)^k$
phi_inv: (x/alpha)^(1/k)+x0$
tra: transform(h,q0,phi,x);
/* [wxMaxima: input   end   ] */



/* Old versions of Maxima abort on loading files that end in a comment. */
"Created with wxMaxima 23.05.1"$