Bessel equation differential equations (Laplace Transform)
peter.vlasschaert@gmail.com,29/09/2017

find : Laplace transform  bessel_j(0,t) with use Bessel differential equation.
info:  https://www.youtube.com/watch?v=II_JUVmSDa8

kill(all)$
/*load('contrib_ode)$*/

'bessel differential equation for n = 0'
==================================
=>       bessel function from order zero

t^2*y''+t*y'+(t^2-n^2)*y = 0  (1)
example : t=x ,y=y(x)

depends(y,x);

q:n=0;

p1:y1=diff(y(x),x);
p2:y2=diff(y(x),x,2);
p3:x^2-n^2;
p33:ev(p3,q);
p4:ev(bessel_j(n,x),q);

suppose x<>0,  use 'p6c' ('Bessel differential equation for  p4 ')

p5:x^2*rhs(p2)+x*rhs(p1)+p33*y (x)= 0;
p6:factor(p5);
p6a:part(p6,1,1);
p6b:part(p6,1,2);
p6c:p6b=0;

1e) Take Laplace Transform from 'p6c'.

p7:laplace(lhs(p6c), x, s);
p8:laplace(y(x),x,s) = ys;
p9:ev(p7,p8)=0;

2e)  How to find ys from 1e order ODE

p10:ode2(p9,ys,s)$
p11:lhs(p10) =ev(rhs(p10),%c=c1);
p12:ratsimp(log(p11));

use  menu : simplify -> expand logarithms

p13:log(ys)=log(c1*%e^(-log(s^2+1)/2)), logexpand=super;

3e) find c1 from 'p11' use    'initial value theorem'.

a)  initial value theorem t=x , s=s

p14:'limit(F(t), t, 0) = 'limit(f(s)*s, s, inf);

first :lhs  (bessel_j(0,0) = 1)

p14:ev(p4,x=0);

second : rhs

p15:limit(s*rhs(p11),s,inf);

p16:p15=p14;

rem : final value theorem

p14:'limit(F(t), t, inf) = 'limit(f(s)*s, s, 0);

b) fill in value c1 in 'p11'

p15:lhs(p11)=ev(rhs(p11),p16);

c) simplify  'p15'

p16:log(p15);
p17:solve(p16,ys)[1];
p18:radcan(p17);

result  : math notation    :Jo(t)
                maxi notation    :bessel_j(n,t)  =>  n=0 => bessel_j(0,t) 
L          = Laplace Transformation
L^(-1)  =Inverse Laplace Transformation
L{f(t)}  = f(s)  <=>  L^(-1){f(s)} = f(t)
-------------------------------------------------------------------------------------------
L{ bessel_j(0,t) } = 1/sqrt(s^2+1)  from 'p18'

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