Luento 16

23.10.2001 HA

Warning, the name changecoords has been redefined

Esimerkki kompleksisista ominaisarvoista

> restart: with(linalg): with(LinearAlgebra):

Warning, the name changecoords has been redefined

Warning, the protected names norm and trace have been redefined and unprotected

Warning, the assigned name GramSchmidt now has a global binding

> A:=<<-1,2>|<-5,-3>>;

> eigenvectors(A);

> om:=Eigenvectors(A);

> lambda:=(om[1])[1];w:=Column(om[2],1);

> alpha:=Re(lambda);beta:=Im(lambda);

> u:=map(Re,w);v:=map(Im,w);

Yleinen ratkaisukaava:

> y:=exp(alpha*t)*((a*cos(beta*t)+b*sin(beta*t))*u+(b*cos(beta*t)-a*sin(beta*t))*v);

> yy:=evalm(%);

> matrix(2,1,yy): Y:=convert(%,Matrix);

> A.Y;

> map(simplify,%);

> opp:=%:

> vp:=map(diff,Y,t):vp:=map(simplify,%);

> vp-opp;

Siis yleinen ratkaisu on oikein.

Alkuehto:

> subs(t=0,Y)=<1,1>;

> C:=<<1/2|3/2>,<1|0>>;b:=<1,1>;

> LinearSolve(C,b);

> y:=subs(a=%[1],b=%[2],y);

> yy:=evalm(%);

Tarkistetaan vielä AE (derivointitarkistus jo suoritettiin).

> subs(t=0,op(yy));map(eval,%);

> subs(t=0,y);map(eval,%);

Jälkimmäisessä ei tarvittu op :ia.

Katsotaan dsolvella:

> dys:=diff(y1(t),t)=-y1(t)-5*y2(t),diff(y2(t),t)=2*y1(t)-3*y2(t);

> AE:=y1(0)=1,y2(0)=1;

> ratk:=dsolve({dys,AE},{y1(t),y2(t)});

> Y1:=subs(ratk,y1(t));Y2:=simplify(subs(ratk,y2(t)));

> op(yy);

Kyllä saatiin samat