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> with(DEtools):with(LinearAlgebra):

> A:=Matrix([ [ 18, 3, 38 ], [ 177, 41, 473 ], [ -19, -4, -47 ] ]);

> M:=Matrix([ [ 1, 2, -2 ], [ 8, 29, -28 ], [ -1, -3, 3 ] ]);

> J:=M^(-1).A.M;

> g:=Matrix([[20*cos(t)^2+12-32*t],[160*cos(t)^2+174-448*t],[-20*cos(t)^2-18+48*t]]);

> h:=M^(-1).g;

> h[3,1];

> de1:=diff(x3(t),t)=4*x3(t)+h[3,1];

> sx3:=solve(dsolve(de1),x3(t));

> de2:=diff(x2(t),t)=4*x2(t)+sx3+6*h[2,1];

> sx2:=solve(dsolve(de2),x2(t));

> de3:=diff(x1(t),t)=4*x1(t)+sx2+20*cos(t)^2;

> sx1:=solve(dsolve(de3),x1(t));

> xv:=Matrix([[sx1],[sx2],[sx3]]);

> xvv:=subs({_C3=0,_C2=0,_C1=0},xv);

> M.xvv;

> A;

> de1:=diff(x1(t),t)=18*x1(t)+3*x2(t)+38*x3(t)+g[1,1];

> de2:=diff(x2(t),t)=177*x1(t)+41*x2(t)+473*x3(t)+g[2,1];

> de3:=diff(x3(t),t)=-19*x1(t)-4*x2(t)-47*x3(t)+g[3,1];

> sol:=dsolve({de1,de2,de3});

$sol := {x1(t) = 2*exp(4*t)*_C2*t+39/4*t+exp(4*t)*_C3-8*exp(4*t)*_C2-exp(4*t)*_C1-2*cos(2*t)+sin(2*t)-37/16-_C2*exp(4*t)*t^2-_C3*exp(4*t)*t, x2(t) = -10*exp(4*t)*_C2*t+139*t-5*exp(4*t)*_C3+38*exp(4*t)*...$ $sol := {x1(t) = 2*exp(4*t)*_C2*t+39/4*t+exp(4*t)*_C3-8*exp(4*t)*_C2-exp(4*t)*_C1-2*cos(2*t)+sin(2*t)-37/16-_C2*exp(4*t)*t^2-_C3*exp(4*t)*t, x2(t) = -10*exp(4*t)*_C2*t+139*t-5*exp(4*t)*_C3+38*exp(4*t)*...$ $sol := {x1(t) = 2*exp(4*t)*_C2*t+39/4*t+exp(4*t)*_C3-8*exp(4*t)*_C2-exp(4*t)*_C1-2*cos(2*t)+sin(2*t)-37/16-_C2*exp(4*t)*t^2-_C3*exp(4*t)*t, x2(t) = -10*exp(4*t)*_C2*t+139*t-5*exp(4*t)*_C3+38*exp(4*t)*...$ $sol := {x1(t) = 2*exp(4*t)*_C2*t+39/4*t+exp(4*t)*_C3-8*exp(4*t)*_C2-exp(4*t)*_C1-2*cos(2*t)+sin(2*t)-37/16-_C2*exp(4*t)*t^2-_C3*exp(4*t)*t, x2(t) = -10*exp(4*t)*_C2*t+139*t-5*exp(4*t)*_C3+38*exp(4*t)*...$ $sol := {x1(t) = 2*exp(4*t)*_C2*t+39/4*t+exp(4*t)*_C3-8*exp(4*t)*_C2-exp(4*t)*_C1-2*cos(2*t)+sin(2*t)-37/16-_C2*exp(4*t)*t^2-_C3*exp(4*t)*t, x2(t) = -10*exp(4*t)*_C2*t+139*t-5*exp(4*t)*_C3+38*exp(4*t)*...$

> subs({_C1=0,_C2=0,_C3=0},sol);

> g;

[ [ 18, 3, 38 ], [ 177, 41, 473 ], [ -19, -4, -47 ] ]