How to solve Bessel equations,Laplace Transform ,tables
peter.vlasschaert@gmail.com,04/10/2017

kill(all)$
depends ([Y],t)$

find    L{Y'(t)} => Y(0)

p1:fs=laplace(Y(t), t, s);

p2:laplace(diff(Y(t),t), t, s);

p3:ratsubst(lhs(p1), rhs(p1), p2);

find L{Y''(t)}

p4:laplace(diff(Y(t),t,2), t, s);

p5:ratsubst(lhs(p1), rhs(p1), p4);

p6:part(p5,1,1);

p7:-p6+part(p5,2)+part(p5,3);

find L^(-1) {s^n} = 0  if n=0,1,2,3,.......

assume(n>=0);

declare(n,integer);

p8:ilt(s^n,s,t)=0;

Bessel ODE 'variable coefficients',initial conditions Y(0), 'p6'

p9:t*diff(Y(t),t,2)+diff(Y(t),t)+4*t*Y(t) = 0;
p9a : Y(0) = 3;
p9b:  p6 = 0;

general :find :L{t^n*f(t)}  <=> (-1)^n*diff(fs,s,n)

find:L{t*f(t)}     <=> -diff(p3,s) , n=1

p10:-4*diff(rhs(p1),s);
p10a:ev(p10,rhs(p1)=lhs(p1));

find:L{t*f'(t)}    <=> -diff(p3,s) , L{f'(t)} = s*f(s)

p11:ev(p3,p9a);

find:L{t*f ''(t)}  <=> -diff(p7,s),n=1

p12:ev(p7,p9a,p9b,p1);
p12a:-diff(p12,s);

take L from =>    t*('diff(Y(t),t,2))+'diff(Y(t),t,1)+4*t*Y(t)=0, s = domain,

p13:p10a+p11+p12a=0;
p14:ev(p13,rhs(p1) = lhs(p1));
p15:ode2(p14,fs,s);
p16:radcan(p15);

go back to time domain ( from : s domain -> t domain) ? No solution

p17:ilt(rhs(p16),s,t);

from tables :  1/(s^2+a^2)  <=> Jo(a*t) = bessel_j(0,2*t) , a=2

a:2$;

p18:yy = %c*bessel_j(0,a*t);

from given : info  Y(0) = 3 , find %c

p19:ev(Y(t) =rhs(p18),t=0);
p20:ev(p19,p9a);

find Y(t) from ODE  'p9' with Maxima ,Tables and %c;

p21: Y(t) = rhs(p18);
p22:lhs(p21) =ev(rhs(p21),rhs(p20)=lhs(p20));