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

appendix: general system 2 differential equations.
[email protected]  ,26/03/2017

(%i3) depends(X,t);
depends(Y,t);
\[\tag{\%{}o2}\label{o2} [\operatorname{X}(t)]\] \[\tag{\%{}o3}\label{o3} [\operatorname{Y}(t)]\]
(%i5) pg1:'diff(X,t)= a*X+b*Y;
pg2:'diff(Y,t)= c*X+d*Y;
\[\tag{pg1}\label{pg1}\frac{d}{dt}X=Yb+Xa\] \[\tag{pg2}\label{pg2}\frac{d}{dt}Y=Yd+Xc\]
(%i10) M1:zeromatrix(2,2);
M1[1,1]:a$
M1[1,2]:b$
M1[2,1]:c$
M1[2,2]:d$
\[\tag{M1}\label{M1}\begin{pmatrix}0 & 0\\ 0 & 0\end{pmatrix}\]

coefficient matrix,pg1,pg2

(%i11) p0:M1;
\[\tag{p0}\label{p0}\begin{pmatrix}a & b\\ c & d\end{pmatrix}\]

general solution' pg1,pg2 '  X=a1*e^(lambda*t)
                            Y=a2*e^(lambda*t)

(%i13) p1:X=a1*%e^(lambda*t);
p2:Y=a2*%e^(lambda*t);
\[\tag{p1}\label{p1}X=\mathit{a1}\,{{\%{}e}^{t\,\mathit{lambda}}}\] \[\tag{p2}\label{p2}Y=\mathit{a2}\,{{\%{}e}^{t\,\mathit{lambda}}}\]
(%i15) p3:ev(pg1,p1,p2);
p4:ev(pg2,p1,p2);
\[\tag{p3}\label{p3}\frac{d}{dt}\left( \mathit{a1}\,{{\%{}e}^{t\,\mathit{lambda}}}\right) =\mathit{a2}b\,{{\%{}e}^{t\,\mathit{lambda}}}+a\,\mathit{a1}\,{{\%{}e}^{t\,\mathit{lambda}}}\] \[\tag{p4}\label{p4}\frac{d}{dt}\left( \mathit{a2}\,{{\%{}e}^{t\,\mathit{lambda}}}\right) =\mathit{a2}d\,{{\%{}e}^{t\,\mathit{lambda}}}+\mathit{a1}c\,{{\%{}e}^{t\,\mathit{lambda}}}\]
(%i17) p5:ev(p3,nouns)/%e^(t*lambda);
p6:ev(p4,nouns)/%e^(t*lambda);
\[\tag{p5}\label{p5}\mathit{a1}\,\mathit{lambda}={{\%{}e}^{-t\,\mathit{lambda}}}\,\left( \mathit{a2}b\,{{\%{}e}^{t\,\mathit{lambda}}}+a\,\mathit{a1}\,{{\%{}e}^{t\,\mathit{lambda}}}\right) \] \[\tag{p6}\label{p6}\mathit{a2}\,\mathit{lambda}={{\%{}e}^{-t\,\mathit{lambda}}}\,\left( \mathit{a2}d\,{{\%{}e}^{t\,\mathit{lambda}}}+\mathit{a1}c\,{{\%{}e}^{t\,\mathit{lambda}}}\right) \]
(%i19) p7:ratsimp(p5);
p8:ratsimp(p6);
\[\tag{p7}\label{p7}\mathit{a1}\,\mathit{lambda}=\mathit{a2}b+a\,\mathit{a1}\] \[\tag{p8}\label{p8}\mathit{a2}\,\mathit{lambda}=\mathit{a2}d+\mathit{a1}c\]
(%i21) p9:rhs(p7)-lhs(p7)=0;
p10:rhs(p8)-lhs(p8)=0;
\[\tag{p9}\label{p9}-\mathit{a1}\,\mathit{lambda}+\mathit{a2}b+a\,\mathit{a1}=0\] \[\tag{p10}\label{p10}-\mathit{a2}\,\mathit{lambda}+\mathit{a2}d+\mathit{a1}c=0\]
(%i25) p11:coeff(part(p9,1),a1);
p12:coeff(part(p9,1),a2);
p13:coeff(part(p10,1),a1);
p14:coeff(part(p10,1),a2);
\[\tag{p11}\label{p11}a-\mathit{lambda}\] \[\tag{p12}\label{p12}b\] \[\tag{p13}\label{p13}c\] \[\tag{p14}\label{p14}d-\mathit{lambda}\]

build matrix M2 ' coefficient matrix '

(%i26) M2:zeromatrix(2,2);
\[\tag{M2}\label{M2}\begin{pmatrix}0 & 0\\ 0 & 0\end{pmatrix}\]
(%i30) M2[1,1]:p11$
M2[1,2]:p12$
M2[2,1]:p13$
M2[2,2]:p14$

coefficient matrix,p9,p10

(%i31) M2;
\[\tag{\%{}o31}\label{o31} \begin{pmatrix}a-\mathit{lambda} & b\\ c & d-\mathit{lambda}\end{pmatrix}\]

solvable : determinant = 0

(%i32) p15: ratsimp(determinant(M2))=0;
\[\tag{p15}\label{p15}{{\mathit{lambda}}^{2}}+\left( -d-a\right) \,\mathit{lambda}+ad-bc=0\]

characteristics polynomial of a matrix  M2,'p15'

(%i33) load("functs")$

needed :tracematrix 'statement'
definition : sum of all elements on diagonal.

(%i34) p16:tracematrix(M1);
\[\tag{p16}\label{p16}d+a\]
(%i35) p17:determinant(M1);
\[\tag{p17}\label{p17}ad-bc\]

lambda^2-tracematrix(M1)*lambda+determinant(M1)=0

(%i36) p18:lambda^2-tracematrix(M1)*lambda+determinant(M1)=0;
\[\tag{p18}\label{p18}{{\mathit{lambda}}^{2}}-\left( d+a\right) \,\mathit{lambda}+ad-bc=0\]

eigenvalues : lambda1,lambda2,see module3
other words : propervalues,char.values

(%i37) p19:solve(p18,lambda)$

two lambda1: 'p20',lambda2: 'p21'

(%i39) p20:p19[1];
p21:p19[2];
\[\tag{p20}\label{p20}\mathit{lambda}=-\frac{\sqrt{{{d}^{2}}-2ad+4bc+{{a}^{2}}}-d-a}{2}\] \[\tag{p21}\label{p21}\mathit{lambda}=\frac{\sqrt{{{d}^{2}}-2ad+4bc+{{a}^{2}}}+d+a}{2}\]

find eigenvectors a1,a2 → lambda1
                 a1,a2 → lambda2

(%i41) p22:a1*M2[1,1]+a2*M2[1,2]=0;
p23:a1*M2[2,1]+a2*M2[2,2]=0;
\[\tag{p22}\label{p22}\mathit{a1}\,\left( a-\mathit{lambda}\right) +\mathit{a2}b=0\] \[\tag{p23}\label{p23}\mathit{a2}\,\left( d-\mathit{lambda}\right) +\mathit{a1}c=0\]

lambda1 find 'p20'

(%i43) p24:ev(p22,p20);
p25:ev(p23,p20);
\[\tag{p24}\label{p24}\mathit{a1}\,\left( \frac{\sqrt{{{d}^{2}}-2ad+4bc+{{a}^{2}}}-d-a}{2}+a\right) +\mathit{a2}b=0\] \[\tag{p25}\label{p25}\mathit{a2}\,\left( \frac{\sqrt{{{d}^{2}}-2ad+4bc+{{a}^{2}}}-d-a}{2}+d\right) +\mathit{a1}c=0\]

solve system 'p24,p25' for a1,a2
menu:equations → solve linear system

(%i44) p26:linsolve([p24, p25], [a1,a2]);
\[\mbox{}\\\mbox{solve: dependent equations eliminated: (2)}\] \[\tag{p26}\label{p26}[\mathit{a1}=\mathit{\%{}r1},\mathit{a2}=-\frac{\mathit{\%{}r1}\,\sqrt{{{d}^{2}}-2ad+4bc+{{a}^{2}}}-\mathit{\%{}r1}d+\mathit{\%{}r1}a}{2b}]\]
(%i45) p27:ev(p26,%r1=1);
\[\tag{p27}\label{p27}[\mathit{a1}=1,\mathit{a2}=-\frac{\sqrt{{{d}^{2}}-2ad+4bc+{{a}^{2}}}-d+a}{2b}]\]

lambda2 find 'p21'

(%i47) p28:ev(p22,p21);
p29:ev(p23,p21);
\[\tag{p28}\label{p28}\mathit{a1}\,\left( a-\frac{\sqrt{{{d}^{2}}-2ad+4bc+{{a}^{2}}}+d+a}{2}\right) +\mathit{a2}b=0\] \[\tag{p29}\label{p29}\mathit{a2}\,\left( d-\frac{\sqrt{{{d}^{2}}-2ad+4bc+{{a}^{2}}}+d+a}{2}\right) +\mathit{a1}c=0\]

solve system 'p24,p25' for a1,a2
menu:equations → solve linear system

(%i48) p30:linsolve([p28, p29], [a1,a2]);
\[\mbox{}\\\mbox{solve: dependent equations eliminated: (2)}\] \[\tag{p30}\label{p30}[\mathit{a1}=\mathit{\%{}r2},\mathit{a2}=\frac{\mathit{\%{}r2}\,\sqrt{{{d}^{2}}-2ad+4bc+{{a}^{2}}}+\mathit{\%{}r2}d-\mathit{\%{}r2}a}{2b}]\]
(%i49) p31:ev(p30,%r2=1);
\[\tag{p31}\label{p31}[\mathit{a1}=1,\mathit{a2}=\frac{\sqrt{{{d}^{2}}-2ad+4bc+{{a}^{2}}}+d-a}{2b}]\]

see: module3,how to write general solution at the 'end'

missing : repeated eigenvalues. 'comming later'
https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/
lecture-26-continuation-repeated-real-eigenvalues/

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