Exercise 2: Inhomogeneous Differential Equation with Initial Conditions
Maple-exercise.
Given the inhomogeneous differential equation
\begin{equation}
x’‘(t)+4x’(t)+29x(t)=-25\sin(2t)+\frac{109}{4}\mathrm e^{-\frac 12 t}-8\cos(2t), \quad t \in \reel.
\end{equation}
D
Find using Maple’s dsolve the complete solution to the differential equation.
E
Plot the solution whose graph passes through the point $(0,1)\,$, and has the slope $-\frac 52\,$ in $\,t=0\,$. Then plot the solution whose graph also passes through the point $(0,1)\,,$ but has the slope $\,\frac 12\,$ i $\,t=0\,$.
hint
See Today’s Maple-Demo.
answer
$ L_{inhom} = \maengde{c_1\e^{-2t}\sin(5t) + c_2\e^{-2t}\cos(5t) + \e^{-\frac{1}{2}t} - \sin(2t)}{t\in\reel,c_1,c_2\in\reel} $
The first of the conditioned solutions: $\, x(t)=\e^{-\frac{t}{2}}-\sin(2t)\,.$
Exercise 3: The Structural Theorem
Consider the linear map $\,f:C^{\infty}(\reel)\rightarrow C^{\infty}(\reel)\,$ given by
$$
f(x(t))=x''(t)+3x'(t)-4x(t)\,.
$$
F
Guess a particular solution to the inhomogeneous differential equation
$$
f(x(t))=29-12t
$$
and then state the complete solution to the equation.
hint
Guess a first-degree polynomial, and then use the result in exercise 1, example 3.
G
Find using the complex guess method a particular solution to the inhomogeneous differential equation
$$
f(x(t))=\cos(t)
$$
and then state the complete solution to the equation.
hint
E.g use a guess of the type $\,x_0(t)=a\,\cos(t)+b\,\sin(t)\,.$
%It applies that Re$(\e^{it})=\cos(t)\,.$ Therefore first guess a function of the type $\,c\e^{it}\,,\,\,c\in \Bbb C,.$
H
Find a particular solution to the inhomogeneous differential equation
$$
f(x(t))=29-12t+\cos(t)$$
and then state the complete real solution to the equation.
is linearly independent (this should not be proved here). In what follows we consider the restriction of $\,f\,$ to the 5-dimensional subspace $\,U\,$ in $\,C^{\infty}(\reel)\,$ that has $\,v\,$ as a basis.
I
Show that the image $\,f(U)\,$ is a subspace in $\,U\,,$ and determine the mapping matrix $\,\matind vFv\,$ for the map $f:U\rightarrow U\,$ with respect to the basis $\,v\,.$
hint
$\,f(U)\,$ is spanned by the images of the basis vectors.
J
Determine the coordinate vector for
$$\,q(t)=\cos(t)+29-12t\,$$
and find all solutions in $\,U\,$ to the equation
$$\,f(x(t))=q(t)\,.$$
K
Does a particular solution $\,x_0(t) \in U\,$ exist to the equation
$$\,f(x(t))=q(t)\,$$
that fulfills the initial conditions $\,x_0(0)=0\,$ and $\,x_0’(0)=1?\,$ Comment!
Exercise 4: Modelling of a Physical Situation
In this week’s eMaple about differential equations you shall model a physical situation using Maple and use the model for experiments. The procedure is that you execute the commands one at a time - so do not use the Maple-key !!! that computes the entire sheet at once. Note the fields XX that you must replace with your own Maple-command. When you have finished an answer, you are welcome to click the triangle for the suggested solution.
A
Download the file eMaple3 and have fun with the modelling!
Exercise 5: Uniqueness of the Solution. Theory
About a differential equation of the form
\begin{equation}
x’‘(t)+a_1x’(t)+a_0x(t)=q(t), \quad t \in \reel
\end{equation}
it is stated that $\,x_1(t)=\sin(t)\,$ and $\,x_2(t)=\frac{1}{2}\sin(2t)\,$ both are solutions.
B
Prove using the existence and uniqueness theorem that the statement is false.
hint
Use Theorem 18.19 in eNote 18.
hint
Do numbers $\,t_0\,$ that fulfill $\,x_{1}(t_0)=x_{2}(t_0)\,$ and $\,x_{1}’(t_0)=x_{2}’(t_0)\,$ exist?
answer
When you have found a number $t_0$ that fulfills the two equations in the last hint, it follows from the
existence and uniqueness theorem that the statement is false. Formulate yourself precisely how!
Exercise 6: Structure of Solutions. Theory
Given the inhomogeneous differential equation
\begin{equation}
x’‘(t)+a_1x’(t)+a_0x(t)=q(t), \quad t \in \reel.
\end{equation}
together with two particular solutions,
\begin{equation}
x_1(t)=\sin t+2\e^t \quad \mathrm{and} \quad x_2(t)=\sin t+\e^t-\e^{-t}.
\end{equation}
A
Determine the complete solution to the homogeneous equation.
hint
How would you utilize the information that the two functions are both a particular solution to the inhomogeneous equation.
hint
The difference between the two functions is a particular solution to the homogeneous equation.
hint
Find $x_1(t)-x_2(t)$. The roots of the characteristic equation are found as the exponents in the function thus found. Moreover this function is a solution to the corresponding homogeneous equation.
Determine the complete solution to the inhomogeneous equation.
C
Determine $\,a_0,\,a_1\,$ and $\,q(t)\,.$
Opg 7: Complex Guess Method
D
Determine a particular solution to the complex differential equation
\begin{equation}
x’‘(t)-2x’(t)-3x(t)=10\,\e^{(-1+2i)t}, \quad t \in \reel.
\end{equation}
hint
Guess a function of the type $\,x_0(t)=c\e^{(-1+2i)t}\,.$
answer
$$\,c=-\frac 12 +i\,.$$
E
Determine a particular solution to the differential equation
\begin{equation}
x’‘(t)-2x’(t)-3x(t)=10\,\e^{-t}\,\cos(2t), \quad t \in \reel.
\end{equation}
Determine a particular solution to the differential equation
\begin{equation}
x’‘(t)-2x’(t)-3x(t)=10\,\e^{-t}\,\sin(2t), \quad t \in \reel.
\end{equation}
Exercise 8: From the Solution to the Differential Equation
All real solutions to an inhomogeneous linear differential equation of second order are
\begin{equation}
x(t)=c_1\e^{-t}\cos 2t+c_2\e^{-t}\sin 2t+5t^3+t^2+12t+7,\quad (c_1,c_2)\in\reel^2.
\end{equation}
G
State the differential equation.
hint
When you have the complete solution to the inhomogeneous equation, you can easily read the complete solution to the corresponding homogeneous equation.
hint
From the complete solution to the corresponding homogeneous equation it is possible to to read the roots to the characteristic equation.
hint
The homogeneous equation is
$$x''(t)+2x'(t)+5x(t)=0\,.$$
You find the inhomogeneous differential equation by ``sending the tail through’’:
$$x_0(t)=5t^3+t^2+12t+7$$
is a solution to the inhomogeneous equations, so the right-hand side of the inhomogeneous equation is exactly
