lecture8.mw

 > with(LinearAlgebra):with(plots):with(DEtools):

Warning, the name changecoords has been redefined

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);

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