\\\\( \nonumber \newcommand{\bevisslut}{$\blacksquare$} \newenvironment{matr}[1]{\hspace{-.8mm}\begin{bmatrix}\hspace{-1mm}\begin{array}{#1}}{\end{array}\hspace{-1mm}\end{bmatrix}\hspace{-.8mm}} \newcommand{\transp}{\hspace{-.6mm}^{\top}} \newcommand{\maengde}[2]{\left\lbrace \hspace{-1mm} \begin{array}{c|c} #1 & #2 \end{array} \hspace{-1mm} \right\rbrace} \newenvironment{eqnalign}[1]{\begin{equation}\begin{array}{#1}}{\end{array}\end{equation}} \newcommand{\eqnl}{} \newcommand{\matind}[3]{{_\mathrm{#1}\mathbf{#2}_\mathrm{#3}}} \newcommand{\vekind}[2]{{_\mathrm{#1}\mathbf{#2}}} \newcommand{\jac}[2]{{\mathrm{Jacobi}_\mathbf{#1} (#2)}} \newcommand{\diver}[2]{{\mathrm{div}\mathbf{#1} (#2)}} \newcommand{\rot}[1]{{\mathbf{rot}\mathbf{(#1)}}} \newcommand{\am}{\mathrm{am}} \newcommand{\gm}{\mathrm{gm}} \newcommand{\E}{\mathrm{E}} \newcommand{\Span}{\mathrm{span}} \newcommand{\mU}{\mathbf{U}} \newcommand{\mA}{\mathbf{A}} \newcommand{\mB}{\mathbf{B}} \newcommand{\mC}{\mathbf{C}} \newcommand{\mD}{\mathbf{D}} \newcommand{\mE}{\mathbf{E}} \newcommand{\mF}{\mathbf{F}} \newcommand{\mK}{\mathbf{K}} \newcommand{\mI}{\mathbf{I}} \newcommand{\mM}{\mathbf{M}} \newcommand{\mN}{\mathbf{N}} \newcommand{\mQ}{\mathbf{Q}} \newcommand{\mT}{\mathbf{T}} \newcommand{\mV}{\mathbf{V}} \newcommand{\mW}{\mathbf{W}} \newcommand{\mX}{\mathbf{X}} \newcommand{\ma}{\mathbf{a}} \newcommand{\mb}{\mathbf{b}} \newcommand{\mc}{\mathbf{c}} \newcommand{\md}{\mathbf{d}} \newcommand{\me}{\mathbf{e}} \newcommand{\mn}{\mathbf{n}} \newcommand{\mr}{\mathbf{r}} \newcommand{\mv}{\mathbf{v}} \newcommand{\mw}{\mathbf{w}} \newcommand{\mx}{\mathbf{x}} \newcommand{\mxb}{\mathbf{x_{bet}}} \newcommand{\my}{\mathbf{y}} \newcommand{\mz}{\mathbf{z}} \newcommand{\reel}{\mathbb{R}} \newcommand{\mL}{\bm{\Lambda}} \newcommand{\mnul}{\mathbf{0}} \newcommand{\trap}[1]{\mathrm{trap}(#1)} \newcommand{\Det}{\operatorname{Det}} \newcommand{\adj}{\operatorname{adj}} \newcommand{\Ar}{\operatorname{Areal}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Rum}{\operatorname{Rum}} \newcommand{\diag}{\operatorname{\bf{diag}}} \newcommand{\bidiag}{\operatorname{\bf{bidiag}}} \newcommand{\spanVec}[1]{\mathrm{span}{#1}} \newcommand{\Div}{\operatorname{Div}} \newcommand{\Rot}{\operatorname{\mathbf{Rot}}} \newcommand{\Jac}{\operatorname{Jacobi}} \newcommand{\Tan}{\operatorname{Tan}} \newcommand{\Ort}{\operatorname{Ort}} \newcommand{\Flux}{\operatorname{Flux}} \newcommand{\Cmass}{\operatorname{Cm}} \newcommand{\Imom}{\operatorname{Im}} \newcommand{\Pmom}{\operatorname{Pm}} \newcommand{\IS}{\operatorname{I}} \newcommand{\IIS}{\operatorname{II}} \newcommand{\IIIS}{\operatorname{III}} \newcommand{\Le}{\operatorname{L}} \newcommand{\app}{\operatorname{app}} \newcommand{\M}{\operatorname{M}} \newcommand{\re}{\mathrm{Re}} \newcommand{\im}{\mathrm{Im}} \newcommand{\compl}{\mathbb{C}} \newcommand{\e}{\mathrm{e}} \\\\)

Exercise 1: Systems of Linear Differential Equations

A

By hand: A system of linear differential equations with constant coefficients is given like this

$$ \begin{matr}{rr} x_1'(t) \\\\ x_2'(t) \end{matr} = \begin{matr}{rr} 1 & 8\\\\ 1 & -1 \end{matr} \begin{matr}{rr} x_1(t) \\\\ x_2(t) \end{matr}, \quad t \in \reel $$

  1. Find the eigenvalues and the corresponding eigenspaces of the system matrix, and - using this - state the complete real solution to the system of differential equations.

  2. Find the solution to the system of differential equations that fulfill $\,x_1(0)=0\,$ and $\,x_2(0)=3\,.$

B

Given the system of differential equations \begin{equation} \begin{matr}{rr} x_1’(t) \\ x_2’(t) \end{matr} = \begin{matr}{rr} 2 & -5\
1 & -2 \end{matr} \begin{matr}{rr} x_1(t) \\ x_2(t) \end{matr}, \quad t \in \reel \end{equation}

  1. Find the eigenvalues and the corresponding eigenspaces of the system matrix, and - using this - state the complete complex solution to the system of differential equations.

  2. Now state the complete real solution to the system of differential equations.

  3. Now we shall find the solution to the system of differential equations that fulfill $\,x_1(0)=0\,$ and $\,x_2(0)=3\,.$ Consider the following: Do we get the same solution if we find the solution by the use of the complete complex solution as by the use of the complete real solution?

C

Given the system of differential equations \begin{equation} \begin{matr}{rr} x_1’(t) \\ x_2’(t) \end{matr} = \begin{matr}{rr} 2 & 1\
-4 & -2 \end{matr} \begin{matr}{rr} x_1(t) \\ x_2(t) \end{matr}, \quad t \in \reel \end{equation}

  1. Find the eigenvalues and the corresponding eigenspaces of the system matrix, and - using this - state the complete real solution to the system of differential equations.

  2. Find the solution to the system of differential equations that fulfill $\,x_1(0)=0\,$ and $\,x_2(0)=3\,.$

%og indsæt deres grafer i samme Maple-plot.

%####### begin:hint %Følg for det første system metoden i \tref{NUID13-eks.diffsys1}{eksempel} eller alternativt i \tref{NUID13-saet.diffsys.strukturloes1}{metode}. Følg for de sidste to systemer metoden i \tref{NUID13-saet.diffsys.strukturloes1}{metode}. %####### end:hint %####### begin:hint %Se \href{http://e-math.imm.dtu.dk/fileadmin/maple/11b_DifflignSystem.mw}{e-maple} om differentialligningssystemer. %####### end:hint

Exercise 2: Homogeneous Linear System of Differential Equations

A homogeneous linear system of differential equations consisting of three equations in the unknown functions $\,x_1(t),x_2(t)\,$ and $\,x_3(t)\,$ (with $t\in\reel$) has the system matrix $\mA$ that has been treated in Maple like this:

3system.png

D

How do the three differential equations look like in their ordinary form (not the matrix-form).

E

State the complete real solution both in matrix form and the ordinary form.

Exercise 3: The Structural Theorem. Theory

Let $\,C^{\infty}(\Bbb R,\Bbb C))\,$ denote the functional vector space of infinitely many times differentiable complex functions of a real variable with $\,\Bbb C\,$ as the corresponding scalar field. For $\,t\in \reel\,$ the functions $\,\cos(t),\,\e^{2it}\,$ and $\,t^3\,$ are examples of vectors in the vector-space. Now we consider the map $\,f:(C^{\infty}(\Bbb R, \Bbb C))^2\rightarrow (C^{\infty}(\Bbb R,\Bbb C))^2\,$ given by

$$ f(\mx(t))=\mx'(t)-\,\mA\mx(t)\,,\,\,\,t \in \reel\,$$

where $\,\mA\,$ is a real $2\times2$-matrix, $\,\mx(t)=(x_1(t),x_2(t))\,$ and $\,\mx’(t)=(x_1’(t),x_2’(t))\,.$

F

Show that $\,f\,$ is linear.

F

$f\,$ must fulfill the two linearity conditions: $f(\mx(t)+\my(t))=f(\mx(t))+f(\my(t))\,.$ $f(k\mx(t))=kf(\mx(t))\,.$

G

Explain that systems of differential equations like the ones that are treated i today’s Exercise 1, can be considered to be homogeneous vector equations of the type

$$\,f(\mx(t))=\mnul\,.$$

H

How can the structural theorem be applied to vector equations of the type

$$\,f(\mx(t))=\mathbf q(t)\,$$

where $\,\mathbf q(t)\in (C^{\infty}(\Bbb R,\Bbb C))^2\,.$ Give a precise formulation.

Exercise 4: The Structural Theorem. By Hand

A

Find the complete solution to the system of differential equations \begin{equation} \begin{matr}{rr} x_1’(t) \\ x_2’(t) \end{matr} = \begin{matr}{rr} 1 & 1\\ 0 & -2 \end{matr}\cdot \begin{matr}{rr} x_1(t) \\ x_2(t) \end{matr}\,\,,\quad t\in \reel\,. \end{equation}

B

Guess a solution to the system of differential equations \begin{equation} \begin{matr}{rr} x_1’(t) \\ x_2’(t) \end{matr} = \begin{matr}{rr} 1 & 1\\ 0 & -2 \end{matr}\cdot \begin{matr}{rr} x_1(t) \\ x_2(t) \end{matr}+\begin{matr}{rr} 0\\ -2t \end{matr}, \quad t\in \reel \end{equation}

and then state the complete real solution to the system of differential equations.

Exercise 5: The Structural Theorem. Maple

A linear system of differential equations are given by \begin{align} \frac{\mathrm d}{\mathrm dx}x_1(t)&=\frac 12x_1(t)-x_2(t)+\cos(4t)\
\frac{\mathrm d}{\mathrm dx}x_2(t)&=\frac 32x_1(t)-2x_2(t)-1 \end{align}

A

State the system of differential equations in matrix form (as in question J in the previous exercise). Find eigenvectors and the corresponding eigenvalues for the system matrix.

B

Find using Maple’s dsolve the complete solution to the system of differential equations. How can the solution be interpreted, on the one hand in the context of eigenvectors and eigenvalues of the system matrix and on the other hand in the context of the structural theorem?

C

Plot the solution that fulfills $\,x_1(0)=10\,$ and $\,x_2(0)=5\,,$ first for $\,t\in \left[\,0,10\,\right]\,$ and then for $\,t\in \left[\,10,20\,\right]\,$ and comment the result.

%### Exercise 6: Modelling of a Physical Situation

%In this exercise you yourself from scratch shall model a physical situation using Maple and make experiments with the model. It gives (together with Exercise 5) a warming up to Theme Exercise 4. Here we consider a homogeneous system, whereas the Theme Exercise concerns inhomogeneous systems. The procedure is that you execute the commands one at a time - so do not use the Maple-button !!! that will execute the entire sheet at once. When you have finished the answer to a question you are welcome to click on the triangle to see the suggested solution.

%####### begin:question %Now download the file eMaple2 %and have fun with the modelling.

%####### end:question

%—-

Exercise 6: Existence and Uniqueness. Advanced.

Again we consider the following system of differential equations from Exercise 1:

$$ \begin{matr}{rr} x_1'(t) \\\\ x_2'(t) \end{matr} = \begin{matr}{rr} 1 & 8\\\\ 1 & -1 \end{matr} \begin{matr}{rr} x_1(t) \\\\ x_2(t) \end{matr}, \quad t \in \reel. $$
A

Explain that for every tuple of numbers $(t_0,a_0,b_0)$ exactly one solution $(x_1(t),x_2(t))$ to the system of differential equations exists such that $x_1(t_0)=a_0$ and $x_2(t_0)=b_0$.

B

Given an arbitrary number tuple $\,(t_0,a_0,b_0)\,$. Consider if there always - for systems of differential equations of the type considered in the exercise above - exists exactly one solution $\,(x_1(t),x_2(t))\,$ to the system of differential equations such that $x_1(t_0)=a_0$ and $x_2(t_0)=b_0$.