lecture8.mw

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

Warning, the name changecoords has been redefined

Separation/Inhomogeneous

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

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

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

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

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

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

> dsolve({de1,de2});

{x2(t) = exp(2*t)*_C2-2*exp(5*t)*_C1-24/5, x1(t) = exp(2*t)*_C2+exp(5*t)*_C1-18/5}

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

>

[Plot]

Inhomogeneous, nonautonomous

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

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

> E:=Eigenvectors(A);

E := Vector[column]([[3], [-1]]), Matrix([[1/2, (-1)/2], [1, 1]])

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

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

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

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

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

Matrix([[1, 1], [4, 1]])

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

g1(t) := exp(2*t)

g2(t) := sin(t)

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

de1 := diff(x1(t), t) = x1(t)+x2(t)+exp(2*t)

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

de2 := diff(x2(t), t) = 4*x1(t)+x2(t)+sin(t)

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

gv := Matrix([[exp(2*t)], [sin(t)]])

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

h := Matrix([[-1/2*exp(2*t)+1/4*sin(t)], [1/2*exp(2*t)+1/4*sin(t)]])

> h[1,1];

-1/2*exp(2*t)+1/4*sin(t)

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

den1 := diff(y1(t), t) = -y1(t)-1/2*exp(2*t)+1/4*sin(t)

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

den2 := diff(y2(t), t) = 3*y2(t)+1/2*exp(2*t)+1/4*sin(t)

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

sol := {y2(t) = -1/2*exp(2*t)-1/40*cos(t)-3/40*sin(t)+exp(3*t)*_C1, y1(t) = -1/6*exp(2*t)-1/8*cos(t)+1/8*sin(t)+exp(-t)*_C2}

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

y1s := 1/24*(-4*exp(2*t)*exp(t)-3*cos(t)*exp(t)+3*sin(t)*exp(t)+24*_C2)/exp(t)

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

y2s := -1/2*exp(2*t)-1/40*cos(t)-3/40*sin(t)+exp(3*t)*_C1

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

xsol := Matrix([[-1/24*(-4*exp(2*t)*exp(t)-3*cos(t)*exp(t)+3*sin(t)*exp(t)+24*_C2)/exp(t)-1/2*exp(2*t)-1/40*cos(t)-3/40*sin(t)+exp(3*t)*_C1], [1/12*(-4*exp(2*t)*exp(t)-3*cos(t)*exp(t)+3*sin(t)*exp(t)+...

> r:=A.xsol+gv;

r := Matrix([[1/24*(-4*exp(2*t)*exp(t)-3*cos(t)*exp(t)+3*sin(t)*exp(t)+24*_C2)/exp(t)-1/2*exp(2*t)-3/40*cos(t)-9/40*sin(t)+3*exp(3*t)*_C1], [-1/12*(-4*exp(2*t)*exp(t)-3*cos(t)*exp(t)+3*sin(t)*exp(t)+2...

> expand(r[1,1]);

-2/3*exp(t)^2-1/5*cos(t)-1/10*sin(t)+_C2/exp(t)+3*exp(t)^3*_C1

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

-2/3*exp(t)^2-1/5*cos(t)-1/10*sin(t)+_C2/exp(t)+3*exp(t)^3*_C1

> expand(r[2,1]);

-8/3*exp(t)^2+1/10*cos(t)+3/10*sin(t)-2*_C2/exp(t)+6*exp(t)^3*_C1

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

-8/3*exp(t)^2+1/10*cos(t)+3/10*sin(t)-2*_C2/exp(t)+6*exp(t)^3*_C1

>

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]]);

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);

c := x^5-12*x^4+57*x^3-134*x^2+156*x-72

> solve(c,x);

3, 3, 2, 2, 2

> 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 := 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;

Matrix([[2, 0, 0, 0, 0], [0, 2, 0, 0, 0], [0, 0, 2, 0, 0], [0, 0, 0, 3, 0], [0, 0, 0, 0, 3]])

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

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

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

h := Matrix([[-9*sin(t)-exp(5*t)+20*exp(2*t)+10*t*exp(2*t)], [18*sin(t)+4*exp(5*t)-39*exp(2*t)-20*t*exp(2*t)], [-45*sin(t)-10*exp(5*t)+100*exp(2*t)+51*t*exp(2*t)], [cos(t)-18*sin(t)-4*exp(5*t)+39*exp(...

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

de1 := diff(y1(t), t) = 2*y1(t)-9*sin(t)-exp(5*t)+20*exp(2*t)+10*t*exp(2*t)

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

de2 := diff(y2(t), t) = 2*y2(t)+18*sin(t)+4*exp(5*t)-39*exp(2*t)-20*t*exp(2*t)

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

de3 := diff(y3(t), t) = 2*y1(t)-45*sin(t)-10*exp(5*t)+100*exp(2*t)+51*t*exp(2*t)

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

de4 := diff(y4(t), t) = 3*y4(t)+cos(t)-18*sin(t)-4*exp(5*t)+39*exp(2*t)+20*t*exp(2*t)

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

de5 := diff(y5(t), t) = 3*y1(t)+45*sin(t)+10*exp(5*t)-100*exp(2*t)-50*t*exp(2*t)

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

sol := {y1(t) = 5*exp(2*t)*t^2+9/5*cos(t)+18/5*sin(t)-1/3*exp(5*t)+20*t*exp(2*t)+exp(2*t)*_C3, y4(t) = -59*exp(2*t)+3/2*cos(t)+11/2*sin(t)-2*exp(5*t)-20*t*exp(2*t)+exp(3*t)*_C4, y2(t) = -10*exp(2*t)*t...sol := {y1(t) = 5*exp(2*t)*t^2+9/5*cos(t)+18/5*sin(t)-1/3*exp(5*t)+20*t*exp(2*t)+exp(2*t)*_C3, y4(t) = -59*exp(2*t)+3/2*cos(t)+11/2*sin(t)-2*exp(5*t)-20*t*exp(2*t)+exp(3*t)*_C4, y2(t) = -10*exp(2*t)*t...sol := {y1(t) = 5*exp(2*t)*t^2+9/5*cos(t)+18/5*sin(t)-1/3*exp(5*t)+20*t*exp(2*t)+exp(2*t)*_C3, y4(t) = -59*exp(2*t)+3/2*cos(t)+11/2*sin(t)-2*exp(5*t)-20*t*exp(2*t)+exp(3*t)*_C4, y2(t) = -10*exp(2*t)*t...sol := {y1(t) = 5*exp(2*t)*t^2+9/5*cos(t)+18/5*sin(t)-1/3*exp(5*t)+20*t*exp(2*t)+exp(2*t)*_C3, y4(t) = -59*exp(2*t)+3/2*cos(t)+11/2*sin(t)-2*exp(5*t)-20*t*exp(2*t)+exp(3*t)*_C4, y2(t) = -10*exp(2*t)*t...sol := {y1(t) = 5*exp(2*t)*t^2+9/5*cos(t)+18/5*sin(t)-1/3*exp(5*t)+20*t*exp(2*t)+exp(2*t)*_C3, y4(t) = -59*exp(2*t)+3/2*cos(t)+11/2*sin(t)-2*exp(5*t)-20*t*exp(2*t)+exp(3*t)*_C4, y2(t) = -10*exp(2*t)*t...

> 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))]]);

yv := Matrix([[5*exp(2*t)*t^2+9/5*cos(t)+18/5*sin(t)-1/3*exp(5*t)+20*t*exp(2*t)+exp(2*t)*_C3], [-10*exp(2*t)*t^2-18/5*cos(t)-36/5*sin(t)+4/3*exp(5*t)-39*t*exp(2*t)+exp(2*t)*_C5], [5*exp(2*t)*t^2+81/2*...

> xsol:=M.yv;

xsol := Matrix([[-10*exp(2*t)*t^2-21/10*cos(t)-17/10*sin(t)-2/3*exp(5*t)-59*t*exp(2*t)+exp(2*t)*_C5-59*exp(2*t)+exp(3*t)*_C4], [1/5*_C1-293/25*sin(t)+107/90*exp(2*t)*_C3+10/9*_C2-293/18*exp(2*t)*t^2+5...xsol := Matrix([[-10*exp(2*t)*t^2-21/10*cos(t)-17/10*sin(t)-2/3*exp(5*t)-59*t*exp(2*t)+exp(2*t)*_C5-59*exp(2*t)+exp(3*t)*_C4], [1/5*_C1-293/25*sin(t)+107/90*exp(2*t)*_C3+10/9*_C2-293/18*exp(2*t)*t^2+5...xsol := Matrix([[-10*exp(2*t)*t^2-21/10*cos(t)-17/10*sin(t)-2/3*exp(5*t)-59*t*exp(2*t)+exp(2*t)*_C5-59*exp(2*t)+exp(3*t)*_C4], [1/5*_C1-293/25*sin(t)+107/90*exp(2*t)*_C3+10/9*_C2-293/18*exp(2*t)*t^2+5...xsol := Matrix([[-10*exp(2*t)*t^2-21/10*cos(t)-17/10*sin(t)-2/3*exp(5*t)-59*t*exp(2*t)+exp(2*t)*_C5-59*exp(2*t)+exp(3*t)*_C4], [1/5*_C1-293/25*sin(t)+107/90*exp(2*t)*_C3+10/9*_C2-293/18*exp(2*t)*t^2+5...xsol := Matrix([[-10*exp(2*t)*t^2-21/10*cos(t)-17/10*sin(t)-2/3*exp(5*t)-59*t*exp(2*t)+exp(2*t)*_C5-59*exp(2*t)+exp(3*t)*_C4], [1/5*_C1-293/25*sin(t)+107/90*exp(2*t)*_C3+10/9*_C2-293/18*exp(2*t)*t^2+5...xsol := Matrix([[-10*exp(2*t)*t^2-21/10*cos(t)-17/10*sin(t)-2/3*exp(5*t)-59*t*exp(2*t)+exp(2*t)*_C5-59*exp(2*t)+exp(3*t)*_C4], [1/5*_C1-293/25*sin(t)+107/90*exp(2*t)*_C3+10/9*_C2-293/18*exp(2*t)*t^2+5...

> r:=A.xsol+gv:

> r[1,1];

21/10*sin(t)-20*exp(2*t)*t^2-17/10*cos(t)-177*exp(2*t)-10/3*exp(5*t)+3*exp(3*t)*_C4-138*t*exp(2*t)+2*exp(2*t)*_C5

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

21/10*sin(t)-20*exp(2*t)*t^2-17/10*cos(t)-177*exp(2*t)-10/3*exp(5*t)+3*exp(3*t)*_C4-138*t*exp(2*t)+2*exp(2*t)*_C5

>

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