For an arbitrary complex number $\,c\neq 0\,$ we consider the differential equation
$$ z'(t)-c\cdot z(t)=2 $$
E
Find using the general formula the complete solution to the differential equation.
hint
You shall probably need Theorem 1.66 and perhaps also the arithmetic rules in Theorem 1.63 in eNote 1.
answer
$z(t)=-\frac 2c + k\,\e^{ct}\,$ where $\,k\in \Bbb C,.$
F
Determine for $\,c=i-1\,$ the conditional solution $\,z(t)\,$ that fulfills $\,z(0)=i\,.$
answer
$z(t)=1+i-\e^{(i-1)t}\,.$
Exercise 2: The Structural Theorem
In this exercise we use knowledge about linear maps to solve three inhomogeneous linear first-order differential equations. In each example we proceed step by step.
G
A mapping $f:C^\infty(\reel)\rightarrow C^\infty(\reel)$ is given by
$$\,f(x(t))= x'(t)\,.$$
Show that $f$ is linear and determine $\,\ker(f)\,.$ State a solution to the equation $\,f(x(t))= \sin (t)\,$ and then state the complete solution to the equation.
hint
The linearity follows from well known rules for the differential quotient, which?
hint
Highschool knowledge: Which functions fulfill that their differential quotients are the constant 0?
hint
State a function whose differential quotient equals $\sin( t)\,.$
answer
$\ker(f)$ are all functions of the type: $x(t)=k\,.$
$x_0(t)=-\cos(t)$ is - as is welll-known - an indefinite integral of $\sin(t)\,.$
The complete solution to $\,f(x(t))= \sin (t)\,$ is according to the structural theorem the functions
$$x(t)=-\cos( t)+k\,,\,\,t \in \reel\,$$
where $k$ is an arbitrary real number.
H
A map $f:C^\infty(\reel)\rightarrow C^\infty(\reel)$ is given by
$$\,f(x(t))= x'(t)-x(t)\,.$$
Show that $f$ is linear and determine $\,\ker(f)\,.$ Guess a solution to the equation $\,f(x(t))= 5\,$ and then state the complete solution to the equation.
hint
Have you forgotten the rules for a linear map? the look at Definition 12.5 in eNote 12.
hint
Highschool knowledge: Which functions fulfill that their differential quotient equals the function itself?
$\ker(f)$ consists of all functions of the type: $x(t)=k\e^t\,.$
$x_0(t)=-5$ is a solution.
The complete solution is according to the structural theorem the functions
$$x(t)=-5+k\e^t\,,\,\,t \in \reel\,$$
where $k$ is an arbitrary real number.
I
A map $f:C^\infty(\reel)\rightarrow C^\infty(\reel)$ is given by
$$f(x(t))= x'(t)+2x(t)\,.$$
Show that $f$ is linear and determine $\,\ker(f)\,.$ Guess a solution to the equation $\,f(x(t))= 2t\,$ and then state the complete solution to the equation.
hint
Again you should be able to find the kernel using your knowledge from highschool.
answer
The complete solution is according to the structural theorem the functions
You should answer this exercise using the structural theorem and superposition.
M
Guess a solution to the differential equation
\begin{equation}
x’(t)+x(t)=2\cos t\,,\,\,t\in\Bbb R.
\end{equation}
hint
Does a solution of the form $x(t)=a \cdot \cos t +b \cdot \sin t$ exist?
hint
Substitute $\,x(t)=a \cdot \cos t+b \cdot \sin t\,$ in the left-hand side of the differential equation and determine $a$ and $b$.
answer
$x(t)= \cos t+ \sin t$
N
Guess a solution to the differential equation
\begin{equation}
x’(t)+x(t)=t^2-1\,,\,\,t\in\Bbb R.
\end{equation}
hint
Substitute $\,x(t)=at^2+bt+c\,$ in the left-hand side of the differential equation and determine $a,$$b$ and $c$.
answer
$x(t)=t^2-2t+1\,.$
O
Solve the differential equation
$$
x'(t)+x(t)=2\cos t +t^2-1\,,\,\,t\in\Bbb R.$$
answer
$x(t)=c\e^{-t}+\cos t+\sin t+t^2-2t+1\,$ where $c$ is an arbitrary real number and $t$ is a real variable.
Exercise 5: Solution and Visualization using Maple
Given the inhomogeneous differential equation
$$ x'(t)+\frac{1}{7}\,x(t)=3-2\cos(t). $$
P
Find using Maple the complete solution fo the differential equation.
hint
Hint to this and the other questions in this exercise: See today’s MapleDemo Basic.
Q
Again find, using Maple, the solution that fulfills the initial condition $ x(0)=0 $.
R
Plot your solution, possibly using different initial conditions.
Exercise 6: Modelling of a Physical Problem
In this exercise we introducere an interactive exercise type called eMaple. The idea is that you yourself right from the start shall model a physical situation using Maple and experiment with the model.
The way to proceed is that you execute commands one at a time - so don’t use the Maple-button !!! that computes the whole sheet at once. After you have completed your answer to a particular question you are welcome to click on the suggested solution.
Exercise 7: Complex Differential Equations. By Hand
The complex functions $\,z(t)\,$ that are defined for $\,t \in \reel\,,$ and that can be differentiated an arbitrary number of times, constitute a vector space which is denoted $\,\left(C^\infty(\reel),\mathbb C\right)\,.$
A linear map $\,f:\left(C^\infty(\reel),\mathbb C\right)\rightarrow \left(C^\infty(\reel),\mathbb C\right)$ is given by
$$f(z(t))= z''(t) + z(t)\,.$$
T
Explain that $\,U=\mathrm{span}\left{\e^{it},\e^{-it}\right}\,$ is a 2-dimensional subspace in $\,\ker(f)\,.$
hint
Show that $\,\e^{it}\,$ og $\,\e^{-it}\,$ both belong to the kernel, and that they are linearly independent.
hint
Linearly independent: Show that
$$\,k_1\e^{it}+k_2\e^{-it}=0\,$$
is only fulfilled for all $\,t\,$ if $\,k_1=k_2=0\,.$ E.g. it shall apply for $\,t=0\,$ and $\,\displaystyle{t=\frac{\pi}2}\,.$
answer
The kernel is (like all kernels) a subspace in the domain. Since $\,\e^{it}\,$ and $\,\e^{-it}\,$ belong to the kernel, every linear kombination of the two vectors must belong to the kernel. Since a spanning is always a subspace and since $\,\e^{it}\,$ and $\,\e^{-it}\,$ are linearly independent, $U$ must be a 2-dimensional subspace in the kernel. (In fact $\,\e^{it}\,$ and $\,\e^{-it}\,$ span the whole kernel, however, we do not yet have sufficient theory to prove this).
U
A real function $\,z_0(t)\,$ in $\,U=\mathrm{span}\left{\e^{it},\e^{-it}\right}$ exists that fulfills the initial value conditions $\,z(0)=1\,$ and $\,z’(0)=0\,.$ Find it!
answer
$$z_0(t)=\cos(t)\,.$$
Exercise 8: Linear mappings between Functional Spaces
Let $\,\mathbf{U}\,$ be the subspace of $\,C^\infty (\reel)\,$ that is spanned by the vectors $\,\cos t$, $\sin t$ and $\e^t\,.$
V
Show that $\,\cos t$, $\sin t$ og $\e^t\,$ constitute a basis for $\,\mathbf{U}\,$
hint
$\mathbf{U}$ is given as the subspace that is spanned by $\cos t$, $\sin t$ and $\e^t$, so we only have to show that the three vectors are linearly independent.
hint
When we shall prove that three vectors, $\mathbf{u}_1$, $\mathbf{u}_2$ and $\mathbf{u}_3$ are linearly independent, it is most easily done by showing that a linear combination of the vectors is only zero, if all coefficients are 0, see Theorem 11.17 in eNote 11.
hint
Can the equation $k_1\cdot\mathbf{u}_1+k_2\cdot\mathbf{u}_2+k_3\cdot\mathbf{u}_3=0\Leftrightarrow k_1\cdot\cos t+k_2\cdot\sin t+k_3\cdot\e^t=0$ have other solutions than the zero solution?
hint
A solution to the equation $k_1\cdot\cos t+k_2\cdot\sin t+k_3\cdot\e^t=0$ must be valid for all $t$. Now try to put $t=0$, $t=\frac{\pi}{2}$ and $t=\pi$ and substitute these values in the equation.
hint
You now get the following three homogeneous equations:
\begin{equation}
\begin{aligned}
k_1\cdot 1+k_2\cdot 0+k_3\cdot 1&=0 \
k_1\cdot 0+k_2\cdot 1+k_3\cdot \e^\frac{\pi}{2}&=0\\
k_1\cdot (-1)+k_2\cdot 0+k_3\cdot \e^\pi &=0.
\end{aligned}
\end{equation}
Solve this system of equations.
hint
First state the augmented matrix corresponding to the system of equations. Does it have full rank?
answer
The equation $k_1\cdot\cos t+k_2\cdot\sin t+k_3\cdot\e^t=0$ is only satisfied for all $t$, if $k_1=k_2=k_3=0$. Therefore the three vectors $\cos t$, $\sin t$ og $\e^t$ are linearly independent and since they span $\mathbf{U}$, they may constitute a basis for $\mathbf{U}$.
A linear map $f:C^\infty (\reel)\rightarrow C^\infty (\reel)$ is given by:
$$f(x(t))= x'(t)+2x(t)\,.$$
W
Show that $f$ maps $\mathbf{U}$ onto itself.
hint
That $f$ maps $\mathbf{U}$ onto it self only means that the image space $f(\mathbf{U})\subseteq\mathbf{U}$.
hint
Therefore we must determine $f(\mathbf{U})$ and show that $f(\mathbf{U})\subseteq\mathbf{U}$.
hint
Since $f$ is linear, $f(\mathbf{U})=\spanVec {f(\cos t), f(\sin t), f(\e^t)}$.
hint
Compute the images of the three basis vectors $f(\cos t)$, $f(\sin t)$ and $f(\e^t)$ and show that they they all belong to $\mathbf{U}$.
answer
The images of the three basis vectors $f(\cos t)=2\cos t -\sin t + 0\cdot\e^t$, $f(\sin t)=\cos t + 2\sin t + 0\cdot\e^t$ and $f(\e^t)=0\cdot\cos t + 0\cdot\sin t + 3e^t$ all belong to $\mathbf{U}$, so $f$ maps $\mathbf{U}$ into itself.
X
State the mapping matrix for $f:\mathbf{U}\rightarrow\mathbf{U}$ with respect to the basis $(\cos t, \sin t, \e^t)$.
hint
The mapping matrix consists of the three images of the three basis vectors expressed as vectors with respect to the basis $(\cos t, \sin t, \e^t)$.
hint
How does the images of the three basis vectors $f(\cos t)=2\cos t -\sin t + 0\cdot\e^t$, $f(\sin t)=\cos t + 2\sin t + 0\cdot\e^t$ and $f(\e^t)=0\cdot\cos t + 0\cdot\sin t + 3e^t$ look, if you express them with respect to the basis $(\cos t, \sin t, \e^t)$?
hint
You only have to state the coefficients in the image as number vectors and gather them in a mapping matrix.
Exercise 9: Linear and Nonlinear Differential Equations
Consider the following seven 1. order differential equations:
$ $$1.\,\,\,\,x’(t)+t \cdot x(t) \cdot (1+x(t))=0, \quad t \in \reel.$$2.\,\,\,\,x’(t)+t^2\cdot x(t)=0, \quad t \in \reel.$$3.\,\,\,\,x’(t)+x(t)=t^2, \quad t \in \reel.$$4.\,\,\,\,x’(t)+(x(t))^2=t, \quad t \in \reel.$$5.\,\,\,\,x’(t)+t^3 \cdot x(t)=0, \quad t \in \reel.$$6.\,\,\,\,x’(t)+\e^{x(t)}=1, \quad t \in \reel.$$7.\,\,\,\,(x’(t))^2+x(t)=0, \quad t \in \reel.$
A
Three of the equations are linear, which?
hint
See Example 16.2 and Example 16.8.
hint
Enter two functions $x_1(t)$ and $x_2(t)$ and their sum $x_1(t)+x_2(t)$ into the left-hand side of the differential equation.
hint
Did the two left-hand sides become equal? Now put $k \cdot x_1(t)$ into the left-hand side. Is it possible to put $k$ outside a bracket?
where $\,vs(x(t))\,$ denotes a function substituted into the left-hand side of the differential equation, the differential equation is linear, otherwise it is nonlinear.
answer
2, 3 and 5 are linear.
B
Solve the three linear equations, using Maple for ‘simulated by hand’.
hint
E.g. use the general solution formula.
answer
$x(t)=c \cdot \e^{-\frac{t^3}{3}}, \quad c \in \reel$
$x(t)=c \cdot \e^{-t}+t^2-2t+2, \quad c \in \reel$
$x(t)=c \cdot \e^{-\frac{t^4}{4}}, \quad c \in \reel$.
C
Find, using Maple, at least one solution to the seven differential equations.
hint
Maple: dsolve.
answer
$x(t) = \dfrac{1}{c \cdot \e^{\frac{t^2}{2}}-1}, \quad c \in \reel$
$x(t)=c \cdot \e^{-\frac{t^3}{3}}, \quad c \in \reel$
$x(t)=c \cdot \e^{-t}+t^2-2t+2, \quad c \in \reel$
$x(t)=\dfrac{c \cdot AiryAi(1,t)+AiryBi(1,t)}{c \cdot AiryAi(t)+AiryBi(t)}, \quad c \in \reel$
$x(t)=c \cdot \e^{-\frac{t^4}{4}}, \quad c \in \reel$
$x(t)=t- \ln (\e^{t+c}-1) +c, \quad c \in \reel$
$x(t)=- \frac{1}{4}t^2+\frac{c}{2}t- \frac{c^2}{4}, \quad c \in \reel$
D
Experiment with the solutions: Plot the solutions for different choices of the arbitrary constant $c$.
