Tagged Questions
1
vote
2answers
62 views
Abstract Geometry?
Are there similar terms in other areas for the idea the "angle" conveys in geometry ? I find that functions for abstract things such as pressure,electrical currents ( nothing geometric there ) on AC ...
4
votes
3answers
102 views
Where to start when learning math (again)?
I have a few questions I hope you can help me answer.
First, I'll introduce myself. I'm a finance undergraduate student in Australia, but I'm originally from Norway. Throughout school I always loved ...
1
vote
2answers
34 views
Solving System of Differential equations
The general solution to differential equation
$$x'=Ax$$ where A is a square matrix is given by solving for the eigenvalues and then eigen vectors of matrix $A$. However, is there a general method if I ...
0
votes
1answer
21 views
What functions are solution to a homogeneous system of differential equations?
Given a vector $\vec{u} \in \mathbb{R}^n$. For what functions $\psi(t)$ can $\vec{x}(t) = \psi(t)\vec{u}$ be a solution of $\dot{\vec{x}} = A \vec{x}$ for some $n \times n$ matrix $A$?
I'm trying to ...
1
vote
0answers
22 views
Derivative methods for artifical neural networks with single hidden layer
I am trying to optimize the output of a given neural network with a single hidden layer. To accomplish this, I intend to find solve for all combinations of inputs where the derivative of the neural ...
2
votes
1answer
44 views
Questions on differential equations of matrices
I have a differential equation $$N'_x(x)=G(x)N(x)$$ where $N, G$ are $2\times2$ matrices depending on $x$, and $G$ satisfies $\sigma G+G\sigma=0$, $\sigma$ is one half of the pauli matrix, i.e. ...
5
votes
1answer
60 views
Having trouble using eigenvectors to solve differential equations
The question asked to solve $$\frac{dx}{dy} = \begin{pmatrix}
5 & 4 \\
-1 & 1\\
\end{pmatrix}x$$ ,where $$ x = \begin{pmatrix} x_1 \\
x_2 \\ \end{pmatrix}$$
I went ...
1
vote
0answers
31 views
A Nonzero Alternating Bilinear Form on the Space $P_1(F)$ Over $F$
Can anybody think of an example of a nonzero alternating bilinear form on the space $P_1(F)$ over $F$.
$F$ is a general field like $\mathbb{R}$ or $\mathbb{C}$.
$P_1(F)$ is the set of all ...
2
votes
0answers
36 views
Orientation-preserving diffeomorphism [duplicate]
Can you help for solving this please. Although I study this subject I could not solve this question please help me ı am willing to learn this question.
1
vote
3answers
38 views
Solution of system of linearly dependent equations.
So, I have the system of equations $x'(t) = Ax$ where $A$ is first row-(4,-2) and second row - (8,-4). This has two eigenvalues, both are 0. But I tried to solve it this way:
$x_1' = 4x_1 -2x_2$
and
...
2
votes
3answers
70 views
Express differential equations as system of first order equations
Express the differential equation $$y'''-6y''-y'+6y=0$$
as a system of first order equations i.e. a matrix equation of the form
$$A(\vec x)'=0$$
where $$\vec x\text{ is the vector }\left[ ...
1
vote
1answer
38 views
3 tank mixing problem
There are 3 tanks filled to capacity with fresh water, all with a 100 liter capacity. At t=0, brine with .5 kg/l salt concentration flows into tank 1 at a 3 l/min rate. The other flows are:
tank 1 -> ...
1
vote
1answer
56 views
Finding three independent solutions for the system
I'm stuck on this assignment. Not sure how to begin.
Let $\begin {bmatrix}2&1&1\\ 0&2&3\\ 0&0&2\end {bmatrix} = A$ as part of the system $Ax=x'$. Find three independent ...
1
vote
1answer
48 views
Differential Equation: Complex Eigenvalue
For the following system
$$ x'=\left( \begin{array}{ccc}
\frac{-1}{2} & 1 \\
-1 & \frac{-1}{2} \end{array} \right)x $$
To find a fundamental set of solutions, we assume that $$ x = Ee^{rt}$$
...
0
votes
0answers
22 views
DFT in complex form and Trigonometric Polynomial Interpolation: why different dimensions of basis vectors set?
I'm trying to figure out, how coefficients of Discrete Fourier Transform in complex form are converted into coefficients of Trigonometric Polynomials Interpolation:
Say, I have a function vector with ...
2
votes
1answer
34 views
Further explanation needed for this first order system of linear equations which is as follows:
I was trying the following problem which was as follows:
Consider the first order system of linear equations:
$\frac{dx}{dt}=AX; \space A=\begin{pmatrix}
3 &2 \\
-2&-1
...
0
votes
2answers
35 views
Linearly independent functions not solutions of ODE
If I have a set of $N$ linearly independent functions $f_1,\dots,f_N$, that may NOT be the solutions of a differential equation, and I impose initial conditions $f(0)=K_0,\dots,D^{N-1}f(0)=K_{N-1}$, ...
3
votes
1answer
57 views
Kernel of Differential Operator
Suppose we have a linear differential equation:
$$\left[D^{n}+a_{1}D^{n-1}+\dots+a_{n}D^{0}\right]y(t)=0$$
Where $y$ is an analytic function.
How can we prove that the kernel of the differential ...
1
vote
0answers
103 views
Kernel of Fractional Differential Operator
Suppose we have a fractional differential equation:
$$\left[D^{nv}+a_{1}D^{\left(n-1\right)v}+\dots+a_{n}D^{0}\right]y(t)=0$$
where $\nu=\frac{1}{q}$ and $q\in\mathbb{N}$ and y is an analytic ...
2
votes
1answer
49 views
which of the following options is correct?
Let $y(t)=\begin{pmatrix}
y_1(t)\\
y_2(t)
\end{pmatrix}$ satisfy $\dfrac {dy}{dt}=Ay; t>0, y(0)=\begin{pmatrix}
0\\
1
\end{pmatrix}$ where $A$ is a $2 \times 2$ constant matrix with real ...
2
votes
1answer
56 views
complex eigenvectors with non zero real parts
I'm wondering about how to deal with complex numbers in eigenvectors that have non zero real parts, as in my eigenvector is $\bigl[\begin{smallmatrix}1-2i\\-1\end{smallmatrix}\bigr]$ that is supposed ...
1
vote
1answer
34 views
Trying to show the differential operator $-e^{-x}\frac{d}{dx}(x\frac{df}{dx})$ is positive
This is a homework question.
Let:
$V$ : $\mathbb{R}$-vector space of $C^\infty$ functions over the interval $[1,a]$ and are $0$ on the boundary of this interval
$\langle-,-\rangle$ : be an inner ...
0
votes
0answers
35 views
Problem in soution similarities
Details on the article Article
From the article equation (9) I got the forth order expansion,
$$ \ddot \phi_4+ \phi_4= a\cos 2 \tau+b \cos \tau+ c \cos (2 \tau-3)+d(\cos2 \tau-3)^2+e \cos^2 \tau+f ...
0
votes
0answers
73 views
Why is there no Overshoot in this second order system?
A second-order system which has a damping factor of less than $1$ should have an overshoot right?
The formula for this is:
$$\exp{(-\pi*\zeta/\sqrt{1-\zeta^2})}$$
But I have noticed that if the ...
0
votes
1answer
49 views
find the general control function
Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference
where:
$A = \begin{pmatrix}
and:
So far I have caculated the controlability matrix to be
$
C
the system is ...
0
votes
1answer
73 views
Construct a matrix transformation, majority of work done need confirmation [duplicate]
Consider
$\frac{dx}{dt} = Ax$ where $A$ is the matrix
$$
\begin{bmatrix}
1 & 0 & 1 \\
0 & 0 & -2 \\
0 & 1 & 0 \\
\end{bmatrix}
$$
...
0
votes
2answers
163 views
Solve the differential equation
Consider
$\frac{dx}{dt} = Ax$ where $A$ is the matrix
$$
\begin{bmatrix}
1 & 0 & 1 \\
0 & 0 & -2 \\
0 & 1 & 0 \\
\end{bmatrix}
$$
...
0
votes
1answer
83 views
Show the stable age structure
Considering the population process described by
where $γ$ is the dominant eigenvalue of $L$ $l$ denotes the survival function of the Leslie matrices and $L$ is the Leslie matrix below
We are trying ...
1
vote
2answers
83 views
Find a general control and then show that this could have been achieved at x2
Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form
$$x_{n+1} = Ax_n + Bu_n,$$
where:
$$A = \begin{pmatrix}
3 & 2 & 2 \\
-1 ...
3
votes
2answers
76 views
Linear ODE repeated eigenvalues how to find more than 2 generalized eigenvectors
So I've searched around the web for a few hours now, as
(i)
$\mathbf A = \begin{pmatrix}2&1\\0&2\end{pmatrix}$
The characteristic polynomial is $(\lambda-2)^2=0$, so $\lambda=2$, repeated.
A ...
0
votes
1answer
60 views
Using centred difference formula to approximate differential equation
(Paper and pencil problem): Let $y_i=y(t_i)$ and $f_i=f(t_i)$, and show that by using the centered difference formula for $y^{\prime\prime}(t)$, we can compute approximations to $y_i$ by ...
0
votes
0answers
34 views
Determine general form of control function andthus show this coul have been achieved earlier [duplicate]
Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form
$$x_{n+1} = Ax_n + Bu_n,$$
where:
$$A = \begin{pmatrix}
3 & 2 & 2 \\
-1 ...
3
votes
1answer
45 views
$y''=y$, $x''=-x$. Write this equation in terms of the first-order system
I'm having trouble with my homework on higher-order equations and their equivalent systems. :(
This is the problem:
Write this equation in terms of the first-order system.
$$\left\{ ...
2
votes
1answer
79 views
meaning of $X'=AX$
i have been trying to learn differential equations and I saw various types and methods to solve those types of equations.
My question is that $X'=AX$ represents linear DE but I am just seeing linear ...
2
votes
2answers
83 views
Determine exponential matrix of A when A has complex eigenvalues
Consider the matrix $A = \begin{pmatrix} 2 & -3 \\ 1 & 4\end{pmatrix}$, which should represent a system of linear differential equations. I need to find the flow $\varphi(t, X)$ whereas $X$ is ...
3
votes
1answer
36 views
Finite family of analytic functions linearly dependent if and only if Wronskian is 0
I know that given two analytic functions on some domain $D$ of the complex plane, then their Wronskian determinant being $0$ is equivalent to them being linearly dependent. I would like to generalise ...
2
votes
1answer
103 views
Exponential of the integral of a matrix - what is wrong with this calculation/statement?
I'm going through a book we're using in our intermediate differential equations course, and this is one of the problems that are contained in it. Note that this question is not tagged as homework, ...
1
vote
0answers
59 views
Invariant Subspaces and Differential Equations
Given
I'm given a marginally stable system, $\dot{x}(t)=Ax(t)$, where$A=\begin{bmatrix} -1 & -10 & -10\cr 1 & 0 & 0\cr 0 & 1 & 0 \end{bmatrix}$, and $x(0)=x_o$.The eigenvalues ...
2
votes
5answers
204 views
Differential equation to Difference equation?
I have the following equation :
$$\frac{dx}{dt} = -5(x-2)$$
$$\frac{dy}{dt} = 0$$
How do I change this differential equation to a difference equation ? Do I use Euler forward method ? I remember ...
0
votes
1answer
69 views
Need to expand $\nabla$ and $\Delta$ included term
In this equation, they used $\nabla$ and $\Delta$, I need to expand them to understand this equation. More to see in this article http://arxiv.org/abs/0802.3525 in equation (15)
\begin{eqnarray}
...
1
vote
1answer
99 views
System of differential equations with triple eigenvalue
Could anybody, please, explain to me, how to solve system of 3 differential equations, when it has triple eigenvalue?
I mean... we solved these equations by creating a matrix $A$ of the system and ...
1
vote
2answers
106 views
general solution to $u'=\left(\begin{matrix} -1 & 1\\ 0 & -1 \end{matrix}\right)u$.
How to find the general solution to the following linear system:
$$\left(\begin{matrix}
u_1'\\
u_2'
\end{matrix}\right)=\left(\begin{matrix}
-1 & 1\\
0 & -1
\end{matrix}\right)
...
1
vote
0answers
41 views
Neglecting solutions and reforming the system of differential equations with reducing the order but to keep choosen solutions
Here I have one problem which should help me to understand how to transform the system of differential equations with the condition to neglect two of four solutions and to get the appropriate system ...
1
vote
2answers
95 views
find all solution to the equation $y'=Ay+b(x)$ for given $A$ and $b(x)$
I am asked to find all solution to the equation:
$$y'= \left( \begin{array}{cc}
13 & 12 \\
12 & 13 \end{array} \right) y+ \left(\begin{array}{c}
x\\
0 \end{array} \right)$$
No initial ...
0
votes
3answers
119 views
Solving system of linear differential equations by eigenvalues
Using eigenvalues and eigenvectors solve system of differential equations:
$$x_1'=x_1+2x_2$$
$$x_2' = 2x_1+x_2$$
And find solution for the initial conditions: $x_1(0) = 1; x_2(0) = -1$
I tried to ...
0
votes
2answers
33 views
Is this an ODE problem, related to linear algebra? May I get a reference text?
The problem:
Let $X = \{ \phi \in C^{2} [0,L] : \phi (0) = \phi (L) = 0 \}$ and we define an operator $T$ on $X$ by $$ T( \phi (x) ) = - \frac{d^{2}}{dx^{2}} \phi (x) = - \phi '' (x). $$ Then
(a) ...
4
votes
3answers
244 views
Differential equation involving matrices
I want to show that
$ Y(t)=\cos(At)$ and $Y(t)=\sin(At)$ satisfy the equation
$$Y''+A^2Y=0 $$
subjected to the initial conditions $Y(0)=I, Y'(0)=0$ and $Y(0)=0, Y'(0)=I$ respictively
where $A$ ...
2
votes
3answers
135 views
How to solve simple systems of differential equations
Say we are given a system of differential equations
$$
\left[ \begin{array}{c} x' \\ y' \end{array} \right] = A\begin{bmatrix} x \\ y \end{bmatrix}
$$
Where $A$ is a $2\times 2$ matrix.
...
1
vote
1answer
103 views
1st-order linear ODE with tridiagonal matrix. Efficient solutions?
I have a 1st-rder linear ODE system where the system is characterized by $A$. Given an initial state $x_0$, I want the state at some later time $t$, efficiently.
$A$ happens to be a symmetric ...
3
votes
1answer
102 views
Outward vectors to an Ellipsoid and Euclidean metrics
I'm reading Arnold's proof of the topologically equivalence of the equations $\dot{x}=Ax$ and $\dot{x}=x$ when all the eigenvalues of the $n \times n$-matrix $A$ have positive real part. The proof is ...
