Bessel equation differential equations : contrib_ode
peter.vlasschaert@gmail.com,28/09/2017

why use for : pde cylindrical coordinates.
pde : partial differential equations.

kill(all)$
load('contrib_ode)$

'bessel differential equation for n = value'
==================================

t^2*y''+t*y'+(t^2-n^2)*y = 0 

depends([y],t);

q:n=1/2;

p1:y1=diff(y,t);
p2:y2=diff(y,t,2);
p3:t^2-n^2;
p33:ev(p3,q);

p3:t^2*rhs(p2)+t*rhs(p1)+p33*y = 0;

p4:contrib_ode(p3,y,t)[1];
p5: method;
p6:ev(p4,%k1=c1,%k2=c2);

=>  check the : 'p6' to put in 'p3'

method 1: answer ?

p7:subst(rhs(p6), y, p3);

p8:ratsimp(ev(p7,diff));

=> build in check method

method 2: answer

 ode_check(p3,p6);

' n ' as integer 'bessel differential equation'

xx^2*yy''+xx*yy'+(xx^2-n^2)*yy = 0 , n=integer

declare(n,integer) ;

depends([yy],xx);

pp1:yy1=diff(yy,xx);
pp2:yy2=diff(yy,xx,2);
pp3:xx^2-n^2;
pp4:xx^2*rhs(pp2)+xx*rhs(pp1)+pp3*yy= 0;
pp5:contrib_ode(pp4,yy,xx)[1];
pp6: method;
pp7:ev(pp5,%k1=cc1,%k2=cc2);

check :result  Bessel Differential Equations for n = integer

pp8:ode_check(pp4,pp7);

info :           /* ...........*/   => removes things from the code