Compute the conditional solution, where $ x(1)=-1$.
answer
$$x(t)=-\frac{t^3}{2}-\frac{1}{2t}$$
For an arbitrary complex number $\,c\neq 0\,$ we consider the differential equation
$$ z'(t)-c\cdot z(t)=2\,. $$
E
Compute using the general formula the general solution to the differential equation.
hint
You will 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 satisfies $\,z(0)=i\,.$
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.
A
A map $f:C^\infty(\reel)\rightarrow C^\infty(\reel)$ is given by
$$\,f(x(t))= x'(t)\,.$$
Show that $f$ is linear and determine $\,\ker(f)\,.$ Compute a solution to the equation $\,f(x(t))= \sin (t)\,$ and then compute the general solution to the equation.
hint
The linearity follows from well-known rules for the derivative. Which?
hint
Which functions have the property that their derivatives are a constant 0?
hint
State a function whose derivative equals $\sin( t)\,.$
answer
$\ker(f)$ consists of all functions of the type: $x(t)=k\,,$ where $k$ is a constant number.
$x_0(t)=-\cos(t)$ is an indefinite integral of $\sin(t)\,.$
The general 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.
B
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 general solution to the equation.
hint
See Definition 12.5 in eNote 12 for a reminder on linear maps.
hint
Which function has the property that it derivative 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 general 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.
C
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 general solution to the equation.
hint
Again you should be able to guess the kernel using knowledge about the derivatives of well-known functions.
answer
The general solution is according to the structural theorem the functions
State in Leibniz notation the three inhomogeneous linear first-order differential equations that are solved above.
answer
The first one can be written as:
$$\frac{\mathrm d}{\mathrm d t}\,x(t)=\sin(t)\,,\,\,t\in\Bbb R$$
Exercise 3: The General Formula or the Structural Theorem?
We are given the differential equation
$$\,\displaystyle{\frac{\mathrm d}{\mathrm d t}\,x(t)+\cos(t)\cdot x(t)=\cos(t)}\,,\,\,t\in\Bbb R.$$
A
Solve the differential equation using the general formula. First, try by hand while using Maple only for the indefinite integrals. Then solve it with Maple’s dsolve.
B
Solve the differential equation using the structural theorem.
hint
First find a solution to the homogeneous differential equation. Then guess a (particular) solution to the inhomogeneous equation. It is very easy!
Solve this exercise using the structural theorem, superposition and the guessing method.
A
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 on 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 in the left-hand side of the differential equation and determine $a$ and $b$.
answer
$x(t)= \cos t+ \sin t$
B
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 in the left-hand side of the differential equation and determine $a,$$b$ and $c$ using the identity theorem for polynomials.
answer
$x(t)=t^2-2t+1\,.$
C
Solve the differential equation
$$
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). $$
A
Compute using Maple’s dsolve the general solution fo the differential equation.
hint
For the questions in this exercise, see today’s Maple demo for how to use the dsolve command and for how to plot your solutions.
B
Again compute using Maple the solution that satisfies the initial condition $x(0)=0$.
C
Plot your solution. Create plots with different initial conditions.
Exercise 6: Modelling of a Physical Problem
In this exercise we will be using an interactive approach that we call eMaple. The idea is that you yourself right from the start will be modelling a physical problem using Maple.
Download the Maple sheet, read the text and execute the commands one at a time (do not use Maple’s execution button !!! that will execute the entire sheet at once). After having solved a question you can open the “Solution” tab to see a suggested solution.
A
Now download the file eMaple1 and carry out the modelling problem.
Exercise 7: Differential Equations for Complex-Valued Functions. By Hand
The complex-valued functions $\,z(t)\,$ that are defined for $\,t \in \reel\,$ and that can be differentiated an arbitrary number of times constitute a vector space which we will denote $\,\left(C^\infty(\reel),\mathbb C\right)\,.$
A linear map $\,f:\left(C^\infty(\reel),\mathbb C\right)\rightarrow \left(C^\infty(\reel),\mathbb C\right)$ is given by
$$f(z(t))= z''(t) + z(t)\,.$$
A
Explain that $\,U=\mathrm{span}\lbrace\e^{it},\e^{-it}\rbrace\,$ is a 2-dimensional subspace of $\,\ker(f)\,.$
hint
Show that $\,\e^{it}\,$ and $\,\e^{-it}\,$ both belong to the kernel, and that they are linearly independent.
hint
To show linear independency in this case, show that the equation
$$\,k_1\e^{it}+k_2\e^{-it}=0\,$$
is only satisfied for all $\,t\,$ if $\,k_1=k_2=0\,.$
hint
That the equation in the previous hint is only satisfied for all $t$ when $k_1=k_2=0$ can be investiated with two examples, such as $\,t=0\,$ and $\,\displaystyle{t=\frac{\pi}2}\,.$
answer
The kernel is (like all kernels) a subspace of the domain. Since $\,\e^{it}\,$ and $\,\e^{-it}\,$ belong to the kernel, every linear kombination of the two vectors must belong to the kernel. Since a span is always a subspace and since $\,\e^{it}\,$ and $\,\e^{-it}\,$ are linearly independent, $U$ must be a 2-dimensional subspace of the kernel. (In fact $\,\e^{it}\,$ and $\,\e^{-it}\,$ span the whole kernel, however, we do not yet have sufficient theory to prove this).
B
A real function $\,z_0(t)\,$ in $\,U=\mathrm{span}\lbrace{\e^{it},\e^{-it}\rbrace}$ satisfies the initial conditions $\,z(0)=1\,$ and $\,z’(0)=0\,.$ Determine this function $z_0(t)$.
answer
$$z_0(t)=\cos(t)$$
Exercise 8: Linear mappings between Functional Spaces
Let $\,\mathbf{U}\,$ be the subspace of $\,C^\infty (\reel)\,$ that is spanned by the vectors $\,\cos (t)$, $\sin (t)$ and $\e^t\,.$
A
Show that $\,\cos (t)$, $\sin (t)$ and $\e^t\,$ constitute a basis for $\,\mathbf{U}\,$
hint
$\mathbf{U}$ is given as the subspace that is spanned by $\cos (t)$, $\sin (t)$ and $\e^t$, so we only have to show that the three vectors are linearly independent.
hint
Proving that three vectors $\mathbf{u}_1$, $\mathbf{u}_2$ and $\mathbf{u}_3$ are linearly independent is most easily done by showing that a linear combination of the vectors is only zero, if all coefficients are 0. See Theorem 11.17 in eNote 11 for a recap.
hint
Can the equation $k_1\cdot\mathbf{u}_1+k_2\cdot\mathbf{u}_2+k_3\cdot\mathbf{u}_3=0\Leftrightarrow k_1\cdot\cos (t)+k_2\cdot\sin (t)+k_3\cdot\e^t=0$ have other solutions than the zero solution?
hint
A solution to the equation $k_1\cdot\cos (t)+k_2\cdot\sin (t)+k_3\cdot\e^t=0$ must be valid for all $t$. Now try to put $t=0$, $t=\frac{\pi}{2}$ and $t=\pi$ and substitute these values into the equation.
hint
You now get the following three homogeneous equations:
\begin{equation}
\begin{aligned}
k_1\cdot 1+k_2\cdot 0+k_3\cdot 1&=0 \newline
k_1\cdot 0+k_2\cdot 1+k_3\cdot \e^\frac{\pi}{2}&=0\newline
k_1\cdot (-1)+k_2\cdot 0+k_3\cdot \e^\pi &=0.
\end{aligned}
\end{equation}
Solve this system of equations.
hint
First state the augmented matrix corresponding to the system of equations. Does it have full rank?
answer
The equation $k_1\cdot\cos (t)+k_2\cdot\sin (t)+k_3\cdot\e^t=0$ is only satisfied for all $t$, if $k_1=k_2=k_3=0$. Therefore the three vectors $\cos (t)$, $\sin (t)$ and $\e^t$ are linearly independent and since they span $\mathbf{U}$, they constitute a basis for $\mathbf{U}$.
A linear map $f:C^\infty (\reel)\rightarrow C^\infty (\reel)$ is given by:
$$f(x(t))= x'(t)+2x(t)\,.$$
B
Show that $f$ maps $\mathbf{U}$ into itself.
hint
That $f$ maps $\mathbf{U}$ into itself means for the image space that $f(\mathbf{U})\subseteq\mathbf{U}$.
hint
Therefore we must determine $f(\mathbf{U})$ and show that $f(\mathbf{U})\subseteq\mathbf{U}$.
hint
Since $f$ is linear, $f(\mathbf{U})=\spanVec \lbracef(\cos t), f(\sin t), f(\e^t)\rbrace$.
hint
Compute the images of the three basis vectors $f(\cos (t))$, $f(\sin (t))$ and $f(\e^t)$ and show that they all belong to $\mathbf{U}$.
answer
The images of the three basis vectors $f(\cos (t))=2\cos (t) -\sin (t) + 0\cdot\e^t$, $f(\sin (t))=\cos (t) + 2\sin (t) + 0\cdot\e^t$ and $f(\e^t)=0\cdot\cos (t) + 0\cdot\sin (t) + 3e^t$ all belong to $\mathbf{U}$, so $f$ maps $\mathbf{U}$ into itself.
C
State the mapping matrix of $f:\mathbf{U}\rightarrow\mathbf{U}$ with respect to the basis $(\cos (t), \sin (t), \e^t)$.
hint
The mapping matrix consists of the three images of the three basis vectors expressed as vectors with respect to the basis $(\cos (t), \sin (t), \e^t)$.
hint
How does the images of the three basis vectors $f(\cos (t))=2\cos (t) -\sin (t) + 0\cdot\e^t$, $f(\sin (t))=\cos (t) + 2\sin (t) + 0\cdot\e^t$ and $f(\e^t)=0\cdot\cos (t) + 0\cdot\sin (t) + 3e^t$ look, if you express them with respect to the basis $(\cos (t), \sin (t), \e^t)$?
hint
You only have to state the coefficients in the image as number vectors and gather them in a mapping matrix.
Exercise 9: Linear and Nonlinear Differential Equations
Consider the following seven 1st-order differential equations:
$ $$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 then 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 substitute in $k \cdot x_1(t)$ into the left-hand side. Is it possible to factorize $k$ outside of 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 solving it “simulated manually”.
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 each of the seven differential equations.
hint
Use Maple’s dsolve.
answer
$x(t) = \dfrac{1}{c \cdot \e^{\frac{t^2}{2}}-1}, \quad c \in \reel$
$x(t)=c \cdot \e^{-\frac{t^3}{3}}, \quad c \in \reel$
$x(t)=c \cdot \e^{-t}+t^2-2t+2, \quad c \in \reel$
$x(t)=\dfrac{c \cdot \mathrm{AiryAi}(1,t)+\mathrm{AiryBi}(1,t)}{c \cdot \mathrm{AiryAi}(t)+\mathrm{AiryBi}(t)}, \quad c \in \reel$
$x(t)=c \cdot \e^{-\frac{t^4}{4}}, \quad c \in \reel$
$x(t)=t- \ln (\e^{t+c}-1) +c, \quad c \in \reel$
$x(t)=- \frac{1}{4}t^2+\frac{c}{2}t- \frac{c^2}{4}, \quad c \in \reel$
D
Experiment with the solutions: Plot the solutions for different choices of the arbitrary constant $c$.
