6.22 problem problem 22

6.22.1 Solution using Matrix exponential method
6.22.2 Solution using explicit Eigenvalue and Eigenvector method

Internal problem ID [379]
Internal file name [OUTPUT/379_Sunday_June_05_2022_01_40_04_AM_84937060/index.tex]

Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number: problem 22.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "system of linear ODEs"

Solve \begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+3 x_{2} \left (t \right )+7 x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-x_{2} \left (t \right )-4 x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=x_{2} \left (t \right )+3 x_{3} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=-6 x_{2} \left (t \right )-14 x_{3} \left (t \right )+x_{4} \left (t \right ) \end {align*}

6.22.1 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_{1}^{\prime }\left (t \right ) \\ x_{2}^{\prime }\left (t \right ) \\ x_{3}^{\prime }\left (t \right ) \\ x_{4}^{\prime }\left (t \right ) \end {array}\right ] &= \left [\begin {array}{cccc} 1 & 3 & 7 & 0 \\ 0 & -1 & -4 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & -6 & -14 & 1 \end {array}\right ]\, \left [\begin {array}{c} x_{1} \left (t \right ) \\ x_{2} \left (t \right ) \\ x_{3} \left (t \right ) \\ x_{4} \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}{cccc} {\mathrm e}^{t} & \frac {{\mathrm e}^{t} t \left (t +6\right )}{2} & {\mathrm e}^{t} t \left (t +7\right ) & 0 \\ 0 & {\mathrm e}^{t} \left (1-2 t \right ) & -4 t \,{\mathrm e}^{t} & 0 \\ 0 & t \,{\mathrm e}^{t} & {\mathrm e}^{t} \left (1+2 t \right ) & 0 \\ 0 & -{\mathrm e}^{t} t \left (t +6\right ) & -2 \,{\mathrm e}^{t} t \left (t +7\right ) & {\mathrm e}^{t} \end {array}\right ] \end {align*}

Therefore the homogeneous solution is \begin {align*} \vec {x}_h(t) &= e^{A t} \vec {c} \\ &= \left [\begin {array}{cccc} {\mathrm e}^{t} & \frac {{\mathrm e}^{t} t \left (t +6\right )}{2} & {\mathrm e}^{t} t \left (t +7\right ) & 0 \\ 0 & {\mathrm e}^{t} \left (1-2 t \right ) & -4 t \,{\mathrm e}^{t} & 0 \\ 0 & t \,{\mathrm e}^{t} & {\mathrm e}^{t} \left (1+2 t \right ) & 0 \\ 0 & -{\mathrm e}^{t} t \left (t +6\right ) & -2 \,{\mathrm e}^{t} t \left (t +7\right ) & {\mathrm e}^{t} \end {array}\right ] \left [\begin {array}{c} c_{1} \\ c_{2} \\ c_{3} \\ c_{4} \end {array}\right ] \\ &= \left [\begin {array}{c} {\mathrm e}^{t} c_{1}+\frac {{\mathrm e}^{t} t \left (t +6\right ) c_{2}}{2}+{\mathrm e}^{t} t \left (t +7\right ) c_{3} \\ {\mathrm e}^{t} \left (1-2 t \right ) c_{2}-4 t \,{\mathrm e}^{t} c_{3} \\ t \,{\mathrm e}^{t} c_{2}+{\mathrm e}^{t} \left (1+2 t \right ) c_{3} \\ -{\mathrm e}^{t} t \left (t +6\right ) c_{2}-2 \,{\mathrm e}^{t} t \left (t +7\right ) c_{3}+{\mathrm e}^{t} c_{4} \end {array}\right ]\\ &= \left [\begin {array}{c} \frac {\left (\left (c_{2}+2 c_{3}\right ) t^{2}+\left (6 c_{2}+14 c_{3}\right ) t +2 c_{1}\right ) {\mathrm e}^{t}}{2} \\ \left (c_{2} \left (1-2 t \right )-4 t c_{3}\right ) {\mathrm e}^{t} \\ {\mathrm e}^{t} \left (t c_{2}+2 t c_{3}+c_{3}\right ) \\ -\left (\left (c_{2}+2 c_{3}\right ) t^{2}+\left (6 c_{2}+14 c_{3}\right ) t -c_{4}\right ) {\mathrm e}^{t} \end {array}\right ] \end {align*}

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

6.22.2 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*}

Or \begin {align*} \left [\begin {array}{c} x_{1}^{\prime }\left (t \right ) \\ x_{2}^{\prime }\left (t \right ) \\ x_{3}^{\prime }\left (t \right ) \\ x_{4}^{\prime }\left (t \right ) \end {array}\right ] &= \left [\begin {array}{cccc} 1 & 3 & 7 & 0 \\ 0 & -1 & -4 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & -6 & -14 & 1 \end {array}\right ]\, \left [\begin {array}{c} x_{1} \left (t \right ) \\ x_{2} \left (t \right ) \\ x_{3} \left (t \right ) \\ x_{4} \left (t \right ) \end {array}\right ] \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}{cccc} 1 & 3 & 7 & 0 \\ 0 & -1 & -4 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & -6 & -14 & 1 \end {array}\right ]-\lambda \left [\begin {array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end {array}\right ]\right ) &= 0 \end {align*}

Therefore \begin {align*} \operatorname {det} \left (\left [\begin {array}{cccc} 1-\lambda & 3 & 7 & 0 \\ 0 & -1-\lambda & -4 & 0 \\ 0 & 1 & 3-\lambda & 0 \\ 0 & -6 & -14 & 1-\lambda \end {array}\right ]\right ) &= 0 \end {align*}

Which gives the characteristic equation \begin {align*} \lambda ^{4}-4 \lambda ^{3}+6 \lambda ^{2}-4 \lambda +1&=0 \end {align*}

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

This table summarises the above result

eigenvalue algebraic multiplicity type of eigenvalue
\(1\) \(1\) real eigenvalue

Now the eigenvector for each eigenvalue are found.

Considering the eigenvalue \(\lambda _{1} = 1\)

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}{cccc} 1 & 3 & 7 & 0 \\ 0 & -1 & -4 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & -6 & -14 & 1 \end {array}\right ] - \left (1\right ) \left [\begin {array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{cccc} 0 & 3 & 7 & 0 \\ 0 & -2 & -4 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & -6 & -14 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}

Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}cccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 0&3&7&0&0\\ 0&-2&-4&0&0\\ 0&1&2&0&0\\ 0&-6&-14&0&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}+\frac {2 R_{1}}{3} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}cccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 0&3&7&0&0\\ 0&0&{\frac {2}{3}}&0&0\\ 0&1&2&0&0\\ 0&-6&-14&0&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {R_{1}}{3} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}cccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 0&3&7&0&0\\ 0&0&{\frac {2}{3}}&0&0\\ 0&0&-{\frac {1}{3}}&0&0\\ 0&-6&-14&0&0 \end {array} \right ] \end {align*}

\begin {align*} R_{4} = R_{4}+2 R_{1} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}cccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 0&3&7&0&0\\ 0&0&{\frac {2}{3}}&0&0\\ 0&0&-{\frac {1}{3}}&0&0\\ 0&0&0&0&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}+\frac {R_{2}}{2} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}cccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 0&3&7&0&0\\ 0&0&{\frac {2}{3}}&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{cccc} 0 & 3 & 7 & 0 \\ 0 & 0 & \frac {2}{3} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{1}, v_{4}\}\) and the leading variables are \(\{v_{2}, v_{3}\}\). Let \(v_{1} = t\). Let \(v_{4} = s\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\{v_{2} = 0, v_{3} = 0\}\)

Hence the solution is \[ \left [\begin {array}{c} t \\ 0 \\ 0 \\ s \end {array}\right ] = \left [\begin {array}{c} t \\ 0 \\ 0 \\ s \end {array}\right ] \] 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} t \\ 0 \\ 0 \\ s \end {array}\right ] &= \left [\begin {array}{c} t \\ 0 \\ 0 \\ 0 \end {array}\right ] + \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ s \end {array}\right ]\\ &= t \left [\begin {array}{c} 1 \\ 0 \\ 0 \\ 0 \end {array}\right ] + s \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 1 \end {array}\right ] \end {align*}

By letting \(t = 1\) and \(s = 1\) then the above becomes \[ \left [\begin {array}{c} t \\ 0 \\ 0 \\ s \end {array}\right ] = \left [\begin {array}{c} 1 \\ 0 \\ 0 \\ 0 \end {array}\right ] + \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 1 \end {array}\right ] \] Hence the two eigenvectors associated with this eigenvalue are \[ \left (\left [\begin {array}{c} 1 \\ 0 \\ 0 \\ 0 \end {array}\right ],\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 1 \end {array}\right ]\right ) \] 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
\(1\) \(4\) \(2\) Yes \(\left [\begin {array}{cc} 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 1 & 0 \end {array}\right ]\)

This case will be solved using the Jordan form of the matrix \(A\). The Jordan form diagonalization is \begin {align*} A &= P J P^{-1} \end {align*}

Which can be found to be \begin {align*} \left [\begin {array}{cccc} 1 & 3 & 7 & 0 \\ 0 & -1 & -4 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & -6 & -14 & 1 \end {array}\right ] &= \left [\begin {array}{cccc} 1 & 3 & 0 & 0 \\ 0 & -2 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ -2 & -6 & 1 & 1 \end {array}\right ] \left [\begin {array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end {array}\right ] \left [\begin {array}{cccc} 1 & 3 & 0 & 0 \\ 0 & -2 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ -2 & -6 & 1 & 1 \end {array}\right ]^{-1} \end {align*}

Looking at the \(P\) matrix above, we see there are \(2\) chains. Therefore, we now construct the basis solution by following these chains as follows. \begin {align*} \vec {x}_1 &= \left [\begin {array}{c} {\mathrm e}^{t} \\ 0 \\ 0 \\ -2 \,{\mathrm e}^{t} \end {array}\right ]\\ \vec {x}_2 &= \left [\begin {array}{c} t \,{\mathrm e}^{t}+3 \,{\mathrm e}^{t} \\ -2 \,{\mathrm e}^{t} \\ {\mathrm e}^{t} \\ -2 t \,{\mathrm e}^{t}-6 \,{\mathrm e}^{t} \end {array}\right ]\\ \vec {x}_3 &= \left [\begin {array}{c} \frac {t^{2} {\mathrm e}^{t}}{2}+3 t \,{\mathrm e}^{t} \\ -2 t \,{\mathrm e}^{t}+{\mathrm e}^{t} \\ t \,{\mathrm e}^{t} \\ -t^{2} {\mathrm e}^{t}-6 t \,{\mathrm e}^{t}+{\mathrm e}^{t} \end {array}\right ]\\ \vec {x}_4 &= \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ {\mathrm e}^{t} \end {array}\right ]\\ \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) + c_{4} \vec {x}_{4}(t) \end {align*}

Which is written as \begin {align*} \left [\begin {array}{c} x_{1} \left (t \right ) \\ x_{2} \left (t \right ) \\ x_{3} \left (t \right ) \\ x_{4} \left (t \right ) \end {array}\right ] &= c_{1} \left [\begin {array}{c} {\mathrm e}^{t} \\ 0 \\ 0 \\ -2 \,{\mathrm e}^{t} \end {array}\right ] + c_{2} \left [\begin {array}{c} t \,{\mathrm e}^{t}+3 \,{\mathrm e}^{t} \\ -2 \,{\mathrm e}^{t} \\ {\mathrm e}^{t} \\ -2 t \,{\mathrm e}^{t}-6 \,{\mathrm e}^{t} \end {array}\right ] + c_{3} \left [\begin {array}{c} \frac {t^{2} {\mathrm e}^{t}}{2}+3 t \,{\mathrm e}^{t} \\ -2 t \,{\mathrm e}^{t}+{\mathrm e}^{t} \\ t \,{\mathrm e}^{t} \\ -t^{2} {\mathrm e}^{t}-6 t \,{\mathrm e}^{t}+{\mathrm e}^{t} \end {array}\right ] + c_{4} \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ {\mathrm e}^{t} \end {array}\right ] \end {align*}

Which becomes \begin {align*} \left [\begin {array}{c} x_{1} \left (t \right ) \\ x_{2} \left (t \right ) \\ x_{3} \left (t \right ) \\ x_{4} \left (t \right ) \end {array}\right ] = \left [\begin {array}{c} \frac {\left (c_{3} t^{2}+\left (2 c_{2}+6 c_{3}\right ) t +2 c_{1}+6 c_{2}\right ) {\mathrm e}^{t}}{2} \\ {\mathrm e}^{t} \left (-2 c_{3} t -2 c_{2}+c_{3}\right ) \\ {\mathrm e}^{t} \left (c_{3} t +c_{2}\right ) \\ -{\mathrm e}^{t} \left (\left (t^{2}+6 t -1\right ) c_{3}+2 t c_{2}+2 c_{1}+6 c_{2}-c_{4}\right ) \end {array}\right ] \end {align*}

Solution by Maple

Time used: 0.031 (sec). Leaf size: 78

dsolve([diff(x__1(t),t)=1*x__1(t)+3*x__2(t)+7*x__3(t)+0*x__4(t),diff(x__2(t),t)=0*x__1(t)-1*x__2(t)-4*x__3(t)+0*x__4(t),diff(x__3(t),t)=0*x__1(t)+1*x__2(t)+3*x__3(t)+0*x__4(t),diff(x__4(t),t)=0*x__1(t)-6*x__2(t)-14*x__3(t)+1*x__4(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \frac {\left (-c_{4} t^{2}-2 c_{3} t -7 c_{4} t +4 c_{2} \right ) {\mathrm e}^{t}}{4} \\ x_{2} \left (t \right ) &= {\mathrm e}^{t} \left (c_{4} t +c_{3} \right ) \\ x_{3} \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (2 c_{4} t +2 c_{3} +c_{4} \right )}{4} \\ x_{4} \left (t \right ) &= \frac {\left (c_{4} t^{2}+2 c_{3} t +7 c_{4} t +2 c_{1} \right ) {\mathrm e}^{t}}{2} \\ \end{align*}

Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 99

DSolve[{x1'[t]==1*x1[t]+3*x2[t]+7*x3[t]+0*x4[t],x2'[t]==0*x1[t]-1*x2[t]-4*x3[t]+0*x4[t],x3'[t]==0*x1[t]+1*x2[t]+3*x3[t]+0*x4[t],x4'[t]==0*x1[t]-6*x2[t]-14*x3[t]+1*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^t (c_2 t (t+6)+2 c_3 t (t+7)+2 c_1) \\ \text {x2}(t)\to -e^t (c_2 (2 t-1)+4 c_3 t) \\ \text {x3}(t)\to e^t ((c_2+2 c_3) t+c_3) \\ \text {x4}(t)\to e^t (c_2 (-t) (t+6)-2 c_3 t (t+7)+c_4) \\ \end{align*}