other method :Laplace Transform L{Jo(t)}
peter.vlasschaert@gmail.com,29/09/2017

How :using series

kill(all);

depends(y,x);

p1:diff(y(x),x);
p2:diff(p1,x);
eq1:x^2*p2+x*p1+(x^2-n^2)*y(x)=0;

declare(n,integer)$

eq2:ode2(eq1,y(x),x);

eq3: ev(yy(x,n)=part(eq2,2),n=0);

eq4:part(eq3,1) =part(eq3,2,2,1);

eq5:subst(part(eq3,2,2,1), y(x), eq1);

eq6:ev(part(eq5,1),diff)=rhs(eq5);

eq7:ev(eq6,n=0);

eq8:radcan(eq7);

check 'eq8' for x ≠ 0, example x=1/2

eq81:ev(eq8,x=1/2);
eq82:float(eq81);

eq9:solve(eq8,x);
eq91:eq9[1];
eq92:eq9[2];

eq10:lhs(eq92)*part(eq92,2,2) = part(eq92,2,1);

serie :  math            notation : Jo(x)
              maxima      notation :  bessel_j(0,x)

eq11:taylor(bessel_j(0,x), x, 0, 10);

cvg  of the serie : Jo(x) = bessel_j(0,x)

eq12:'limit(bessel_j(0,x),x,inf) = limit(bessel_j(0,x),x,inf) ;

find x=0 the serie

eq13:'limit(bessel_j(0,x),x,0) = limit(bessel_j(0,x),x,0) ;

take : Laplace Transform from each term from the serie
=> normal serie go to infinity.

eq14:laplace(eq11,x,s);

use : first ratsubst => factor

eq15:ratsubst(q, 1/s, eq14);
eq16:factor(eq15);
eq161:-1;
eq162:part(eq16,1,1);
eq163:part(eq16,1,2);
eq164:part(eq162,1);
eq165:part(eq162,2);
eq17:ratsimp(eq161*eq165);
eq18:ratsimp(eq17/eq163);
eq19:factorsum(ratsubst(1/s, q, eq18));
ratdenomdivide:true$
eq20:ratexpand(eq19);
eq21:1/s*eq20;
eq211:radcan(eq21);

from other method we find :1/sqrt(s^2+1) = 1/s*(1+1/s^2)^(-1/2) 
rem : from q -> 1/s. ('eq24')

eq22:taylor((1+(s)^2)^(-1/2), s, 0, 10);
eq23:ratsubst(s, 1/s, eq22);
eq24:1/s*eq23;

rem : construct : (1+(1/s)^2)^(-1/2 )  :build with 'eq22' change s by 1/s  and get 'eq23'
rem :  eq23 and eq24 must be infinite serie.