<sagecell> A=matrix(2,2,[0,1,-2,-3])
icy=5 icv=0
t = var('t') y = function('y', t) v = function('v', t)
de1 = diff(y,t) == A[0,0]*y+A[0,1]*v de2 = diff(v,t) == A[1,0]*y+A[1,1]*v
gensol = desolve_system([de1, de2], [y,v])
ivpsol = desolve_system([de1, de2], [y,v], ics=[0,icy,icv]);
table([ ["General Solution",gensol], ["Particular Solution",ivpsol], ["Eigenvalues",A.eigenvalues()], ["Eigenvectors",A.eigenvectors_right()], ])
soln1, soln2 = ivpsol[0].rhs(), ivpsol[1].rhs() p1=plot(soln1,(t,0,1),color='red',legend_label=r"$y(t)$") p1+=plot(soln2,(t,0,1),color='blue',legend_label=r"$v(t)$") show(p1)
p2=parametric_plot((soln1,soln2),(t,0,10),legend_label=r"$(y(t),v(t))$") show(p2)
F(y,v)=(A[0,0]*y+A[0,1]*v, A[1,0]*y+A[1,1]*v) p3=plot_vector_field(F(y,v),(y,-5,5),(v,-5,5)) show(p3)
p4=p2+p3 show(p4)
</sagecell>
You can also download a Mathematica Notebook that will do the same thing. Sometimes Sage is down. If it is, please open this Mathematica Notebook. Remember, if you area student at BYU-Idaho, then Mathematica is free.