Separation/Inhomogeneous

>	A:=Matrix([[3,-1],[-2,4]]);

>	de1:=diff(x1(t),t)=3*x1(t)-x2(t)+6;

>	de2:=diff(x2(t),t)=-2*x1(t)+4*x2(t)+12;

>	dsolve({de1,de2});

>	dfieldplot([de1,de2],[x1(t),x2(t)],t=0..1,x1=-8..2,x2=-8..2,arrows=MEDIUM);

>

Inhomogeneous, nonautonomous

>	A:=Matrix([[1,1],[4,1]]);

>	E:=Eigenvectors(A);

>	M:=Matrix([[-1,1],[2,2]]);

>	C:=Matrix([[-1,0],[0,3]]);

>	M.C.M^(-1);

>	g1(t):=exp(2*t);g2(t):=sin(t);

>	de1:=diff(x1(t),t)=x1(t)+x2(t)+g1(t);

>	de2:=diff(x2(t),t)=4*x1(t)+x2(t)+g2(t);

>	gv:=Matrix([[g1(t)],[g2(t)]]);

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

>

h[1,1];

>	den1:=diff(y1(t),t)=-y1(t)+h[1,1];

>	den2:=diff(y2(t),t)=3*y2(t)+h[2,1];

>	sol:=dsolve({den1,den2});

>	y1s:=solve(sol[2],y1(t));

>	y2s:=solve(sol[1],y2(t));

>	xsol:=M.Matrix([[y1s],[y2s]]);

>	r:=A.xsol+gv;

>	expand(r[1,1]);

>	expand(diff(xsol[1,1],t));

>	expand(r[2,1]);

>	expand(diff(xsol[2,1],t));

>

Higher Dimensional Eigenspaces

>	A:=Matrix([[3,-18,-4,39,20],[0,11,2,-20,-10],[0,9,4,-20,-10],[0,-18,-4,42,20],[0,45,10,-100,-48]]);

>	c:=CharacteristicPolynomial(A,x);

>	solve(c,x);

>	M:=Matrix([[0,1,0,1,0],[-2/9,20/9,10/9,0,1/5],[1,0,0,0,1/5],[0,1,0,0,-2/5],[0,0,1,0,1]]);

>	M^(-1).A.M;

>	gv:=Matrix([[cos(t)],[sin(t)],[exp(5*t)],[exp(2*t)],[t*exp(2*t)]]);

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

>	de1:=diff(y1(t),t)=2*y1(t)+h[1,1];

>	de2:=diff(y2(t),t)=2*y2(t)+h[2,1];

>	de3:=diff(y3(t),t)=2*y1(t)+h[3,1];

>	de4:=diff(y4(t),t)=3*y4(t)+h[4,1];

>	de5:=diff(y5(t),t)=3*y1(t)+h[5,1];

>	sol:=dsolve({de1,de2,de3,de4,de5});

>	yv:=Matrix([[solve(sol[1],y1(t))],[solve(sol[3],y2(t))],[solve(sol[5],y3(t))],[solve(sol[2],y4(t))],[solve(sol[4],y5(t))]]);

>	xsol:=M.yv;

>	r:=A.xsol+gv:

>

r[1,1];

>	diff(xsol[1,1],t);

>

>

>