\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)
(%i1) kill(all)$
(%i1) load(eigen)$

module 2: solve system : differential equations. 2 by 2
[email protected] , 25/03/2017

(%i3) ps[1]:'diff(x[1],t)=-0.5*x[1]+x[2];
ps[2]:'diff(x[2],t)=0*x[1]-2*x[2];
\[\tag{\%{}o2}\label{o2} \frac{d}{dt}{{x}_{1}}={{x}_{2}}-0.5{{x}_{1}}\] \[\tag{\%{}o3}\label{o3} \frac{d}{dt}{{x}_{2}}=-2{{x}_{2}}\]
(%i4) M1:zeromatrix(2,2);
\[\tag{M1}\label{M1}\begin{pmatrix}0 & 0\\ 0 & 0\end{pmatrix}\]

find coef ps[1] ⇒ x1,x2

(%i6) p1:coeff(part(ps[1],2),x[1]);
p2:coeff(part(ps[1],2),x[2]);
\[\tag{p1}\label{p1}-0.5\] \[\tag{p2}\label{p2}1\]
(%i7) for i:1 thru 2 do
   for j:1 thru 2 do
 M1[i,j]:coeff(part(ps[i],2),x[j]);;
\[\tag{\%{}o7}\label{o7} \mathit{done}\]

coefficient matrix (from differential equations): ps[1],ps[2].

(%i8) M1;
\[\tag{\%{}o8}\label{o8} \begin{pmatrix}-0.5 & 1\\ 0 & -2\end{pmatrix}\]
(%i9) [vals, vecs] : uniteigenvectors (M1);
\[\mbox{}\\\mbox{rat: replaced -0.5 by -1/2 = -0.5}\] \[\tag{\%{}o9}\label{o9} [[[-\frac{1}{2},-2],[1,1]],[[[1,0]],[[\frac{2}{\sqrt{13}},-\frac{3}{\sqrt{13}}]]]]\]

⇒ eigenvalues :

(%i10) p3:vals;
\[\tag{p3}\label{p3}[[-\frac{1}{2},-2],[1,1]]\]

values :

(%i12) p4:part(vals,1)[1];
p5:part(vals,1)[2];
\[\tag{p4}\label{p4}-\frac{1}{2}\] \[\tag{p5}\label{p5}-2\]

multiplicities 'values':

(%i14) p41:part(vals,2)[1];
p51:part(vals,2)[2];
\[\tag{p41}\label{p41}1\] \[\tag{p51}\label{p51}1\]

⇒ uniteigenvectors :

(%i16) p6:part(vecs,1)[1];
p7:part(vecs,2)[1];
\[\tag{p6}\label{p6}[1,0]\] \[\tag{p7}\label{p7}[\frac{2}{\sqrt{13}},-\frac{3}{\sqrt{13}}]\]

⇒ build matrix 'eigenvectors,column format'
math : A

(%i17) p8:transpose(matrix(p6,p7));
\[\tag{p8}\label{p8}\begin{pmatrix}1 & \frac{2}{\sqrt{13}}\\ 0 & -\frac{3}{\sqrt{13}}\end{pmatrix}\]

⇒ build from p8 ⇒ 'invert(p8)'
math : A^(-1)

(%i18) p9:invert(p8);
\[\tag{p9}\label{p9}\begin{pmatrix}1 & \frac{2}{3}\\ 0 & -\frac{\sqrt{13}}{3}\end{pmatrix}\]
(%i19) p10:diagmatrix(2,1);
\[\tag{p10}\label{p10}\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}\]
(%i20) for i:1 thru 2 do
   p10[i][i]:e^((part(vals,1)[i])*t);
\[\tag{\%{}o20}\label{o20} \mathit{done}\]

Matrix exponential '2 by 2'
math : e^(At),A = matrix 'diagonal matrix'

(%i21) p10;
\[\tag{\%{}o21}\label{o21} \begin{pmatrix}\frac{1}{{{e}^{\frac{t}{2}}}} & 0\\ 0 & \frac{1}{{{e}^{2t}}}\end{pmatrix}\]

product : three matrix multiplied
math : A*e^(At)*A^(-1)

(%i22) p12:float(p8*p10*p9);
\[\tag{p12}\label{p12}\begin{pmatrix}\frac{1}{{{e}^{\frac{t}{2}}}} & 0.0\\ 0.0 & \frac{1}{{{e}^{2t}}}\end{pmatrix}\]

intial values : need for IVP 'see module 1'

(%i25) x0[1,1]:1;
x0[1,2]:0;
p13:transpose(genmatrix(x0,1,2,1,1))$
\[\tag{\%{}o23}\label{o23} 1\] \[\tag{\%{}o24}\label{o24} 0\]

intial value (IVP): vector

(%i26) p13;
\[\tag{\%{}o26}\label{o26} \begin{pmatrix}1\\ 0\end{pmatrix}\]

solution : system of differential equations.(ps[1],ps[2])
math: A*e^(At)*A^(-1)*x0 , x0 = 'vector'

(%i27) p14:p12.p13;
\[\tag{p14}\label{p14}\begin{pmatrix}\frac{1}{{{e}^{\frac{t}{2}}}}\\ 0.0\end{pmatrix}\]
(%i29) p15:x[1]= p14[1,1];
p16:x[2]= p14[2,1];
\[\tag{p15}\label{p15}{{x}_{1}}=\frac{1}{{{e}^{\frac{t}{2}}}}\] \[\tag{p16}\label{p16}{{x}_{2}}=0.0\]

check : first differential equation :  ps(1)

(%i30) p17:ev('diff(part(p15,2),t)=part(p16,2)-0.5*part(p15,2),nouns);
\[\tag{p17}\label{p17}-\frac{\log{(e)}}{2{{e}^{\frac{t}{2}}}}=-\frac{0.5}{{{e}^{\frac{t}{2}}}}\]

how to bring in e to %e =2.7....

(%i31) p18:float(-log(%e)/2);
\[\tag{p18}\label{p18}-0.5\]
(%i32) p18:ev(p17,e=%e);
\[\tag{p18}\label{p18}-\frac{{{\%{}e}^{-\frac{t}{2}}}}{2}=-0.5{{\%{}e}^{-\frac{t}{2}}}\]

rem : '.' used multiply matrix with vector

module 2:

statement : zeromatrix,genmatrix,diagmatrix,transpose,uniteigenvectors
           ,invert,nouns,coeff.

Created with wxMaxima.