problem 782

Internal problem ID [15512]
Internal ﬁle name [OUTPUT/15513_Friday_May_10_2024_05_47_32_PM_35823396/index.tex]

Book: A book of problems in ordinary diﬀerential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 3 (Systems of diﬀerential equations). Section 20. The method of elimination. Exercises page 212
Problem number: 782.
ODE order: 1.
ODE degree: 1.

Solve \begin {align*} x^{\prime }\left (t \right )&=y \left (t \right )+z \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+z \left (t \right )\\ z^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right ) \end {align*}

27.7.1 Solution using Matrix exponential method

In this method, we will assume we have found the matrix exponential \(e^{A t}\) allready. There are diﬀerent methods to determine this but will not be shown here. This is a system of linear ODE’s given as \begin {align*} \vec {x}'(t) &= A\, \vec {x}(t) \end {align*}

Or \begin {align*} \left [\begin {array}{c} x^{\prime }\left (t \right ) \\ y^{\prime }\left (t \right ) \\ z^{\prime }\left (t \right ) \end {array}\right ] &= \left [\begin {array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end {array}\right ]\, \left [\begin {array}{c} x \left (t \right ) \\ y \left (t \right ) \\ z \left (t \right ) \end {array}\right ] \end {align*}

For the above matrix \(A\), the matrix exponential can be found to be \begin {align*} e^{A t} &= \left [\begin {array}{ccc} \frac {2 \,{\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{2 t}}{3} & \frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3} & \frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3} \\ \frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3} & \frac {2 \,{\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{2 t}}{3} & \frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3} \\ \frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3} & \frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3} & \frac {2 \,{\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{2 t}}{3} \end {array}\right ] \end {align*}

Therefore the homogeneous solution is \begin {align*} \vec {x}_h(t) &= e^{A t} \vec {c} \\ &= \left [\begin {array}{ccc} \frac {2 \,{\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{2 t}}{3} & \frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3} & \frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3} \\ \frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3} & \frac {2 \,{\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{2 t}}{3} & \frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3} \\ \frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3} & \frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3} & \frac {2 \,{\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{2 t}}{3} \end {array}\right ] \left [\begin {array}{c} c_{1} \\ c_{2} \\ c_{3} \end {array}\right ] \\ &= \left [\begin {array}{c} \left (\frac {2 \,{\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{2 t}}{3}\right ) c_{1}+\left (\frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3}\right ) c_{2}+\left (\frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3}\right ) c_{3} \\ \left (\frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3}\right ) c_{1}+\left (\frac {2 \,{\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{2 t}}{3}\right ) c_{2}+\left (\frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3}\right ) c_{3} \\ \left (\frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3}\right ) c_{1}+\left (\frac {{\mathrm e}^{2 t}}{3}-\frac {{\mathrm e}^{-t}}{3}\right ) c_{2}+\left (\frac {2 \,{\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{2 t}}{3}\right ) c_{3} \end {array}\right ]\\ &= \left [\begin {array}{c} \frac {\left (2 c_{1}-c_{2}-c_{3}\right ) {\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{2 t} \left (c_{2}+c_{1}+c_{3}\right )}{3} \\ \frac {\left (-c_{1}+2 c_{2}-c_{3}\right ) {\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{2 t} \left (c_{2}+c_{1}+c_{3}\right )}{3} \\ \frac {\left (-c_{1}-c_{2}+2 c_{3}\right ) {\mathrm e}^{-t}}{3}+\frac {{\mathrm e}^{2 t} \left (c_{2}+c_{1}+c_{3}\right )}{3} \end {array}\right ] \end {align*}

Since no forcing function is given, then the ﬁnal solution is \(\vec {x}_h(t)\) above.

27.7.2 Solution using explicit Eigenvalue and Eigenvector method

This is a system of linear ODE’s given as \begin {align*} \vec {x}'(t) &= A\, \vec {x}(t) \end {align*}

The ﬁrst step is ﬁnd the homogeneous solution. We start by ﬁnding the eigenvalues of \(A\). This is done by solving the following equation for the eigenvalues \(\lambda \) \begin {align*} \operatorname {det} \left ( A- \lambda I \right ) &= 0 \end {align*}

Expanding gives \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end {array}\right ]-\lambda \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) &= 0 \end {align*}

Therefore \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccc} -\lambda & 1 & 1 \\ 1 & -\lambda & 1 \\ 1 & 1 & -\lambda \end {array}\right ]\right ) &= 0 \end {align*}

Which gives the characteristic equation \begin {align*} \lambda ^{3}-3 \lambda -2&=0 \end {align*}

The roots of the above are the eigenvalues. \begin {align*} \lambda _1 &= 2\\ \lambda _2 &= -1 \end {align*}


eigenvalue	algebraic multiplicity	type of eigenvalue

\(-1\)	\(1\)	real eigenvalue

\(2\)	\(1\)	real eigenvalue

We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end {array}\right ] - \left (-1\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}

Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1&1&1&0\\ 1&1&1&0\\ 1&1&1&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-R_{1} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1&1&1&0\\ 0&0&0&0\\ 1&1&1&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-R_{1} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1&1&1&0\\ 0&0&0&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{2}, v_{3}\}\) and the leading variables are \(\{v_{1}\}\). Let \(v_{2} = t\). Let \(v_{3} = s\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\{v_{1} = -t -s\}\)

Hence the solution is \[ \left [\begin {array}{c} v_{1} \\ t \\ s \end {array}\right ] = \left [\begin {array}{c} -t -s \\ t \\ s \end {array}\right ] \] Since there are two free Variable, we have found two eigenvectors associated with this eigenvalue. The above can be written as \begin {align*} \left [\begin {array}{c} v_{1} \\ t \\ s \end {array}\right ] &= \left [\begin {array}{c} -t \\ t \\ 0 \end {array}\right ] + \left [\begin {array}{c} -s \\ 0 \\ s \end {array}\right ]\\ &= t \left [\begin {array}{c} -1 \\ 1 \\ 0 \end {array}\right ] + s \left [\begin {array}{c} -1 \\ 0 \\ 1 \end {array}\right ] \end {align*}

By letting \(t = 1\) and \(s = 1\) then the above becomes \[ \left [\begin {array}{c} v_{1} \\ t \\ s \end {array}\right ] = \left [\begin {array}{c} -1 \\ 1 \\ 0 \end {array}\right ] + \left [\begin {array}{c} -1 \\ 0 \\ 1 \end {array}\right ] \] Hence the two eigenvectors associated with this eigenvalue are \[ \left (\left [\begin {array}{c} -1 \\ 1 \\ 0 \end {array}\right ],\left [\begin {array}{c} -1 \\ 0 \\ 1 \end {array}\right ]\right ) \] Considering the eigenvalue \(\lambda _{2} = 2\)

We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end {array}\right ] - \left (2\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} -2 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 1 & -2 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}

Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -2&1&1&0\\ 1&-2&1&0\\ 1&1&-2&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}+\frac {R_{1}}{2} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -2&1&1&0\\ 0&-{\frac {3}{2}}&{\frac {3}{2}}&0\\ 1&1&-2&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}+\frac {R_{1}}{2} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -2&1&1&0\\ 0&-{\frac {3}{2}}&{\frac {3}{2}}&0\\ 0&{\frac {3}{2}}&-{\frac {3}{2}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}+R_{2} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -2&1&1&0\\ 0&-{\frac {3}{2}}&{\frac {3}{2}}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} -2 & 1 & 1 \\ 0 & -\frac {3}{2} & \frac {3}{2} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{3}\}\) and the leading variables are \(\{v_{1}, v_{2}\}\). Let \(v_{3} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\{v_{1} = t, v_{2} = t\}\)

Hence the solution is \[ \left [\begin {array}{c} v_{1} \\ v_{2} \\ t \end {array}\right ] = \left [\begin {array}{c} t \\ t \\ t \end {array}\right ] \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} v_{1} \\ v_{2} \\ t \end {array}\right ] = t \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ] \] Let \(t = 1\) the eigenvector becomes \[ \left [\begin {array}{c} v_{1} \\ v_{2} \\ t \end {array}\right ] = \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ] \] The following table gives a summary of this result. It shows for each eigenvalue the algebraic multiplicity \(m\), and its geometric multiplicity \(k\) and the eigenvectors associated with the eigenvalue. If \(m>k\) then the eigenvalue is defective which means the number of normal linearly independent eigenvectors associated with this eigenvalue (called the geometric multiplicity \(k\)) does not equal the algebraic multiplicity \(m\), and we need to determine an additional \(m-k\) generalized eigenvectors for this eigenvalue.


	multiplicity

eigenvalue	algebraic \(m\)	geometric \(k\)	defective?	eigenvectors

\(2\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ]\)

\(-1\)	\(2\)	\(2\)	No	\(\left [\begin {array}{cc} -1 & -1 \\ 0 & 1 \\ 1 & 0 \end {array}\right ]\)

Now that we found the eigenvalues and associated eigenvectors, we will go over each eigenvalue and generate the solution basis. The only problem we need to take care of is if the eigenvalue is defective. Since eigenvalue \(2\) is real and distinct then the corresponding eigenvector solution is \begin {align*} \vec {x}_{1}(t) &= \vec {v}_{1} e^{2 t}\\ &= \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ] e^{2 t} \end {align*}

eigenvalue \(-1\) is real and repated eigenvalue of multiplicity \(2\).There are two possible cases that can happen. This is illustrated in this diagram

This eigenvalue has algebraic multiplicity of \(2\) which is the same as its geometric multiplicity \(2\), then it is complete eigenvalue and this falls into case \(1\) shown above. Hence the corresponding eigenvector basis are \begin {align*} \vec {x}_{2}(t) &= \vec {v}_{2} e^{-t}\\ &= \left [\begin {array}{c} -1 \\ 0 \\ 1 \end {array}\right ] e^{-t} \end {align*}

\begin {align*} \vec {x}_{3}(t) &= \vec {v}_{3} e^{-t}\\ &= \left [\begin {array}{c} -1 \\ 1 \\ 0 \end {array}\right ] e^{-t} \end {align*}

Therefore the ﬁnal solution is \begin {align*} \vec {x}_h(t) &= c_{1} \vec {x}_{1}(t) + c_{2} \vec {x}_{2}(t) + c_{3} \vec {x}_{3}(t) \end {align*}

Which is written as \begin {align*} \left [\begin {array}{c} x \left (t \right ) \\ y \left (t \right ) \\ z \left (t \right ) \end {array}\right ] &= c_{1} \left [\begin {array}{c} {\mathrm e}^{2 t} \\ {\mathrm e}^{2 t} \\ {\mathrm e}^{2 t} \end {array}\right ] + c_{2} \left [\begin {array}{c} -{\mathrm e}^{-t} \\ 0 \\ {\mathrm e}^{-t} \end {array}\right ] + c_{3} \left [\begin {array}{c} -{\mathrm e}^{-t} \\ {\mathrm e}^{-t} \\ 0 \end {array}\right ] \end {align*}

Which becomes \begin {align*} \left [\begin {array}{c} x \left (t \right ) \\ y \left (t \right ) \\ z \left (t \right ) \end {array}\right ] = \left [\begin {array}{c} \left (-c_{2}-c_{3}\right ) {\mathrm e}^{-t}+c_{1} {\mathrm e}^{2 t} \\ c_{1} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{-t} \\ c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{-t} \end {array}\right ] \end {align*}

27.7.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [x^{\prime }\left (t \right )=y \left (t \right )+z \left (t \right ), y^{\prime }\left (t \right )=x \left (t \right )+z \left (t \right ), z^{\prime }\left (t \right )=x \left (t \right )+y \left (t \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}\left (t \right )=\left [\begin {array}{c} x \left (t \right ) \\ y \left (t \right ) \\ z \left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Convert system into a vector equation}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}^{\prime }\left (t \right )=\left [\begin {array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{x}}\left (t \right )+\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}^{\prime }\left (t \right )=\left [\begin {array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{x}}\left (t \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}^{\prime }\left (t \right )=A \cdot {\moverset {\rightarrow }{x}}\left (t \right ) \\ \bullet & {} & \textrm {To solve the system, find the eigenvalues and eigenvectors of}\hspace {3pt} A \\ \bullet & {} & \textrm {Eigenpairs of}\hspace {3pt} A \\ {} & {} & \left [\left [-1, \left [\begin {array}{c} -1 \\ 0 \\ 1 \end {array}\right ]\right ], \left [-1, \left [\begin {array}{c} -1 \\ 1 \\ 0 \end {array}\right ]\right ], \left [2, \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair, with eigenvalue of algebraic multiplicity 2}\hspace {3pt} \\ {} & {} & \left [-1, \left [\begin {array}{c} -1 \\ 0 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {First solution from eigenvalue}\hspace {3pt} -1 \\ {} & {} & {\moverset {\rightarrow }{x}}_{1}\left (t \right )={\mathrm e}^{-t}\cdot \left [\begin {array}{c} -1 \\ 0 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Form of the 2nd homogeneous solution where}\hspace {3pt} {\moverset {\rightarrow }{p}}\hspace {3pt}\textrm {is to be solved for,}\hspace {3pt} \lambda =-1\hspace {3pt}\textrm {is the eigenvalue, and}\hspace {3pt} {\moverset {\rightarrow }{v}}\hspace {3pt}\textrm {is the eigenvector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}_{2}\left (t \right )={\mathrm e}^{\lambda t} \left (t {\moverset {\rightarrow }{v}}+{\moverset {\rightarrow }{p}}\right ) \\ \bullet & {} & \textrm {Note that the}\hspace {3pt} t \hspace {3pt}\textrm {multiplying}\hspace {3pt} {\moverset {\rightarrow }{v}}\hspace {3pt}\textrm {makes this solution linearly independent to the 1st solution obtained from}\hspace {3pt} \lambda =-1 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} {\moverset {\rightarrow }{x}}_{2}\left (t \right )\hspace {3pt}\textrm {into the homogeneous system}\hspace {3pt} \\ {} & {} & \lambda \,{\mathrm e}^{\lambda t} \left (t {\moverset {\rightarrow }{v}}+{\moverset {\rightarrow }{p}}\right )+{\mathrm e}^{\lambda t} {\moverset {\rightarrow }{v}}=\left ({\mathrm e}^{\lambda t} A \right )\cdot \left (t {\moverset {\rightarrow }{v}}+{\moverset {\rightarrow }{p}}\right ) \\ \bullet & {} & \textrm {Use the fact that}\hspace {3pt} {\moverset {\rightarrow }{v}}\hspace {3pt}\textrm {is an eigenvector of}\hspace {3pt} A \\ {} & {} & \lambda \,{\mathrm e}^{\lambda t} \left (t {\moverset {\rightarrow }{v}}+{\moverset {\rightarrow }{p}}\right )+{\mathrm e}^{\lambda t} {\moverset {\rightarrow }{v}}={\mathrm e}^{\lambda t} \left (\lambda t {\moverset {\rightarrow }{v}}+A \cdot {\moverset {\rightarrow }{p}}\right ) \\ \bullet & {} & \textrm {Simplify equation}\hspace {3pt} \\ {} & {} & \lambda {\moverset {\rightarrow }{p}}+{\moverset {\rightarrow }{v}}=A \cdot {\moverset {\rightarrow }{p}} \\ \bullet & {} & \textrm {Make use of the identity matrix}\hspace {3pt} \mathrm {I} \\ {} & {} & \left (\lambda \cdot I \right )\cdot {\moverset {\rightarrow }{p}}+{\moverset {\rightarrow }{v}}=A \cdot {\moverset {\rightarrow }{p}} \\ \bullet & {} & \textrm {Condition}\hspace {3pt} {\moverset {\rightarrow }{p}}\hspace {3pt}\textrm {must meet for}\hspace {3pt} {\moverset {\rightarrow }{x}}_{2}\left (t \right )\hspace {3pt}\textrm {to be a solution to the homogeneous system}\hspace {3pt} \\ {} & {} & \left (A -\lambda \cdot I \right )\cdot {\moverset {\rightarrow }{p}}={\moverset {\rightarrow }{v}} \\ \bullet & {} & \textrm {Choose}\hspace {3pt} {\moverset {\rightarrow }{p}}\hspace {3pt}\textrm {to use in the second solution to the homogeneous system from eigenvalue}\hspace {3pt} -1 \\ {} & {} & \left (\left [\begin {array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end {array}\right ]-\left (-1\right )\cdot \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right )\cdot {\moverset {\rightarrow }{p}}=\left [\begin {array}{c} -1 \\ 0 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Choice of}\hspace {3pt} {\moverset {\rightarrow }{p}} \\ {} & {} & {\moverset {\rightarrow }{p}}=\left [\begin {array}{c} -1 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {Second solution from eigenvalue}\hspace {3pt} -1 \\ {} & {} & {\moverset {\rightarrow }{x}}_{2}\left (t \right )={\mathrm e}^{-t}\cdot \left (t \cdot \left [\begin {array}{c} -1 \\ 0 \\ 1 \end {array}\right ]+\left [\begin {array}{c} -1 \\ 0 \\ 0 \end {array}\right ]\right ) \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [2, \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}_{3}={\mathrm e}^{2 t}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {General solution to the system of ODEs}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}=c_{1} {\moverset {\rightarrow }{x}}_{1}\left (t \right )+c_{2} {\moverset {\rightarrow }{x}}_{2}\left (t \right )+c_{3} {\moverset {\rightarrow }{x}}_{3} \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}=c_{1} {\mathrm e}^{-t}\cdot \left [\begin {array}{c} -1 \\ 0 \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{-t}\cdot \left (t \cdot \left [\begin {array}{c} -1 \\ 0 \\ 1 \end {array}\right ]+\left [\begin {array}{c} -1 \\ 0 \\ 0 \end {array}\right ]\right )+c_{3} {\mathrm e}^{2 t}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Substitute in vector of dependent variables}\hspace {3pt} \\ {} & {} & \left [\begin {array}{c} x \left (t \right ) \\ y \left (t \right ) \\ z \left (t \right ) \end {array}\right ]=\left [\begin {array}{c} \left (\left (-t -1\right ) c_{2} -c_{1} \right ) {\mathrm e}^{-t}+c_{3} {\mathrm e}^{2 t} \\ c_{3} {\mathrm e}^{2 t} \\ \left (c_{2} t +c_{1} \right ) {\mathrm e}^{-t}+c_{3} {\mathrm e}^{2 t} \end {array}\right ] \\ \bullet & {} & \textrm {Solution to the system of ODEs}\hspace {3pt} \\ {} & {} & \left \{x \left (t \right )=\left (\left (-t -1\right ) c_{2} -c_{1} \right ) {\mathrm e}^{-t}+c_{3} {\mathrm e}^{2 t}, y \left (t \right )=c_{3} {\mathrm e}^{2 t}, z \left (t \right )=\left (c_{2} t +c_{1} \right ) {\mathrm e}^{-t}+c_{3} {\mathrm e}^{2 t}\right \} \end {array} \]

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{-t} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{-t}+{\mathrm e}^{-t} c_{1} \\ z \left (t \right ) &= c_{2} {\mathrm e}^{2 t}-2 c_{3} {\mathrm e}^{-t}-{\mathrm e}^{-t} c_{1} \\ \end{align*}

\begin{align*} x(t)\to \frac {1}{3} e^{-t} \left (c_1 \left (e^{3 t}+2\right )+(c_2+c_3) \left (e^{3 t}-1\right )\right ) \\ y(t)\to \frac {1}{3} e^{-t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (e^{3 t}+2\right )+c_3 \left (e^{3 t}-1\right )\right ) \\ z(t)\to \frac {1}{3} e^{-t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (e^{3 t}-1\right )+c_3 \left (e^{3 t}+2\right )\right ) \\ \end{align*}

27.7 problem 782

27.7.1 Solution using Matrix exponential method

27.7.2 Solution using explicit Eigenvalue and Eigenvector method

27.7.3 Maple step by step solution