How to solve IVP with unit_step construction with (L transformation),version 1.0.
peter.vlasschaert@gmail.com,09/10/2017                  "wxMaxima 17.10"

kill(all)$
depends ([Y],t)$
depends([G],t)$
load('contrib_ode)$
load(orthopoly)$

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)}, Y(0) = 0, Y'(0) = 0

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);

input system : H(t) = 1-unit_step(t-%pi) = unit_step(t)-unit_step(t-%pi)

p8:wxplot2d([unit_step(t)-unit_step(t-%pi)],[t,-2,5], [y,-3,3])$

p9:unit_step(t)-unit_step(t-%pi);

laplace transform H(t) <=> H(s)

p10:laplace(p9, t, s);

a:2$
b:5$

ODE (IVP) problem: Y(0) = 0,Y'(0)=0  (p6) , H(t)
'diff(Y(t),t,2)+2*('diff(Y(t),t,1))+5*Y(t)=unit_step(t)-unit_step(t-%pi)

p11:diff(Y(t),t,2)+a*diff(Y(t),t)+b*Y(t) = p9;
p12: Y(0) = 0;
p13:  p6 = 0;

laplace transformation  from the above ODE(IVP)

p14:ev(p14:p7+a*p3+b*fs=p10,p12,p13);

solve :fs : Y(s)

p15:factor(solve(p14,fs))[1];

find : inverse Laplace transform  => L^(-1){fs} = Y(t)
rem :  L^(-1) 'linear operator'
=>  L^(-1){a1*f(t)+a2*g(t)} = a1*L^(-1){f(t)}+a2*L^(-1){g(t)}

p16:expand(part(p15,2,1))$
p17:part(p15,2,2)$

fs1:part(p16,1)/p17;
fs2:part(p16,2)/p17;

1e part fs => fs1  => L^(-1){fs1}   => ft1.

ft1:ilt(1/(s*(s^2+2*s+5)), s, t);

2e part  fs => fs3 =>L^(-1){fs2}   => ft2.

F1:L^(-1){exp(-k*s)*F(s)} = f(t-k)*unit_step(t-k)
k : nonnegative constant (example k =3.14159 = pi)

a) ft1 : t  =>  ftk : t-k ( t => t -k)

ftk1:subst(t-k, t, ft1);
ftk:ev(ftk1,k=%pi);

b)ft2: ftk*unit_step(t-pi)

ft2:ftk*unit_step(t-%pi);

find : Y(t)

p18:ft=ft1-ft2;

p19:part(p18,1)=trigsimp(part(p18,2));

check  y(0) = 0 => ft(0)=0

p20::part(p18,1)=ev(part(p19,2),t=0);