1
vote
0answers
16 views

Prove that solutions to linear system form a vector space of dimension $\geq 2$

I accept & appreciate any form of help with the following problem: $B_{nxn}$ "periodic matrix" with period $T$ such that $B(t+T) = B(t)$ for all $t\in \mathbb{R}$. Assume that the system $x' = ...
1
vote
0answers
10 views

Limit of solution of linear system of ODEs as $t\to \infty$

I am completely stuck on the following problem: Consider the linear system: $x'(t)=A(t)x(t)$ where $A(t)$ is an $n$ by $n$ matrix. Assume that $\lim_{t\to \infty}A(t)=B$. Suppose that each eigenvalue ...
0
votes
1answer
21 views

Elementary row operations in matrices

This is really such a lovely math community, I am working on some differential equations hw and my teacher didn't teach this topic yet so I am a little confused. My first question is pertaining to ...
0
votes
0answers
19 views

vector space differential equations

Hi! I am working on some differential equations homework and we are up to the linear algebra part. This particular homework set on Vector space is due, but my teacher has not taught the material yet ...
2
votes
1answer
41 views

A real matrix whose eigenvalues have all negative real parts

While taking a look in some lecture notes of an ODE course, I found the following claim, which appeared in the text as an exercise: Let $A$ be a real $n\times n$ matrix whose eigenvalues have all ...
0
votes
0answers
13 views

Discrete Fourier vectors are the eigenvectors for any linear, constant coefficient, periodic, finite difference discretization on a uniform grid?

I came across the following statement: It can be shown that the DF vectors are always the complete set of eigenvectors of any linear, constant coefficient, periodic, finite difference discretization ...
1
vote
0answers
27 views

Differential equations, derivative of determinant, Euler's formula

Let $b:\mathbb{R}^n\to\mathbb{R}^n$ be a smooth vector field. Let $u(s,x,t):\mathbb{R}^{n+2}\to\mathbb{R}^n$ with $s,t\in\mathbb{R}$ and $x\in\mathbb{R}^n$ satisfy the following differential ...
0
votes
0answers
16 views

final value theorem in the presence of white noise

I apply the final value theorem to get the steady-state error with the presence of white noise and I just keep getting zero. To me, it seems wrong to have zero steady-state error when there is noise ...
0
votes
1answer
20 views

How to prove $Y_1, Y_2, Y_3$ form a fundamental set of solutions to Linear System $Y' = AY$ when eigenvalues of $A$ are defective

Sorry for the long prose. I am trying to understand a naive treatment of the solution to the Linear System with constant coefficients $$ \left({ \begin{matrix} y_1(t) \\ y_2(t) ...
0
votes
1answer
47 views

The finite-dimensional function spaces that are closed under taking derivative

I have been stuck on this problem, I don't know where to start. The exact question is: Determine the finite-dimensional spaces $W$ of differentiable functions with this property: If $f$ is in $W$ then ...
6
votes
1answer
212 views

Recursion relation of fourth order Runge-Kutta method applied on system

I'm trying to apply the Gauss-Legendre method of fourth order (as Runge-Kutta method) on the following system of equations $$\left\{ \begin{matrix} \dot{a} =& -b \\ ...
0
votes
1answer
66 views

Critical Points of Quadratic Forms

just a question about finding critical points (points where the differential is not surjective). I have the equation $$ f(x) = x^tAx $$ where $A$ is a symmetric $n$ by $n$ matrix and $x$ is an element ...
3
votes
5answers
51 views

Linear independence of the functions $1,\cos(x),\cos(2x)$

I want to show that the functions $1,\cos(x),\cos(2x)$ are linearly independent in $C[-\pi,\pi]$. I computed the Wronskian determinat of these functions but at the points $x=0,-\pi,\pi$ the obtained ...
3
votes
3answers
170 views

Why do the concepts of linear algebra apply to differential equations?

A lot of the stuff we do to solve diff equations are taken word for word from linear algebra. The concept of linear independence, determinant of the Wronskian used to determine independence, adding a ...
0
votes
2answers
33 views

References for Linear Algebra needed for Differential Equations and Linear Programming

I am in need of learning the Linear Algebraic theory behind the following Applied disciplines. Could someone please recommend Linear Algebra books for: Differential Equations: Specifically learning ...
8
votes
4answers
525 views

How to explain to a high school student why a linear differential equation is linear?

My mother is teaching a high school course on multivariable calculus, and they were studying linear differential equations of the form $$y' + P(x) y = Q(x),$$ and the question of why this equation is ...
2
votes
3answers
67 views

If $A$ is a $2\times 2$ matrix with a repeated eigenvalue $r$, then $\mathrm{e}^{At}=\mathrm{e}^{rt}\left[I+(A-rI)t\right]$

If $A$ is a $2\times 2$ matrix with a repeated eigenvalue $r$, show that $\mathrm{e}^{At}=\mathrm{e}^{rt}\left[I+(A-rI)t\right]$. I have already been able to show that if $A$ is an arbitrary ...
6
votes
1answer
158 views

Finite dimensional spaces

What are the finite-dimensional spaces $W$ of differentiable functions with this property: If $f$ is in $W$, then $\frac{df}{dx}$ is in $W$.
2
votes
1answer
140 views

Find $f$, such that $\,f,f',\dots,f^{(n-1)}\,$ linearly independent and $\,f^{(n)}=f$

I am trying to find a function $f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{C})$, satisfying the differential equation $$ f^{(n)}=f, $$ and with $\,f,f',\dots,f^{(n-1)}\,$ being linearly independent. ...
0
votes
0answers
29 views

Determinants in pairs of fundamental solutions to particular types of linear, time-varying ODEs

Consider a vector-valued ODE of the following form $$ x'(t) = \begin{bmatrix} 0 & A(t) \\ B(t) & 0 \end{bmatrix}x(t) = \Xi(t) x(t), $$ where $x(t) \in \mathbb{R}^{2n}$ and $A$ and $B$ are ...
0
votes
2answers
26 views

Find solutions for an differential equation system

I have a differential equation system $x_1'(t) = -x_2(t)$ $x_2'(t) = -x_1(t)$ I see that I can write $\dot{x} = Ax$ where $A = \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}$ The complete ...
2
votes
1answer
26 views

Obtain a special solution for a differential equation

If I have a differential equation $y" + 4y' + 4y = 2e^{-x} + xe^{-x}$ and $f(x)$ is a special solution to this equation, how can I determine $f(x)$ if I know f(0)=0 and $f'(0)=0$ ?
1
vote
1answer
35 views

Find $a_1, a_2$ and $F(x)$ for the given differential equation using the special solutions.

Suppose $y_1(x)= e^{-2x} + xe^{-x}$ , $y_2(x)= xe^{-2x} + xe^{-x}$ , $y_3(x)= e^{-2x} - xe^{-2x} + xe^{-x}$ are three special solutions to the differential equation, $y'' + a_1y' + a_2y = F(x)$ , ...
0
votes
0answers
33 views

Quaternion Integration - Initial value problem

We have a standard form of quaternion integration equation $$ q(t) = q(t_0) \exp\left(\frac 12 \int_{t_0}^t \mathbf{\omega}(\tau) d\tau\right),\tag 1 $$ For reference you can check equation (42) in ...
0
votes
0answers
31 views

Quaternion Equivalence

Assume $R_{3\times3}$ is a rotation matrix. Question Is it true that there exists two quaternions representing this same rotation matrix $R_{3\times3}$ ? Hint : $\theta = \arccos\left( ...
0
votes
0answers
26 views

Solving an equation with boundary conditions to find coefficients

I want to find the unknown constants in the function $f(x,y)=A(e^{-i.k_{x}x}+C_{1}x+C_{2})(e^{-i.k_{y}y}+C_{3}y+C_{4})$, using the following known boundary conditions and auxiliary equation ...
0
votes
1answer
65 views

Particular solution of a system of linear differential equations

Let $A(t) \in \mathbb R^{2\times 2}$ and $b(t) \in \mathbb R^2$ continuous functions in an open interval $I$. Consider the system $$(1) \space X'=A(t)X+b(t).$$ Let $X_1,X_2$ be linearly independent ...
2
votes
1answer
56 views

Matrix with eigenvalues no negatives: What is $\lim_{t\to\infty} e^{tA}$?

Here's a homework question I've been stuck on for a while. My question is what can you tell about $$\lim_{t\rightarrow\infty}e^{tA}$$ if $A$ is $n\times n$ matrix and you know that every eigenvalue of ...
2
votes
1answer
34 views

When is the solution to a n initial value problem matrix differential equation invertible?

Suppose $A (t,s)$ a $n\times n$ matrix is the solution of the initial value problem below, where $B_s$ is also an $n\times n$ matrix, invertible for all $s$: $$\dfrac{d A(t,s)}{ds} = B_s A(t,s)$$ $$ ...
0
votes
0answers
26 views

Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
3
votes
1answer
61 views

Showing $y_1$ or $y_2$ are not polynomials

proof that $y_1$ or $y_2$ are not a polynomial for any $n$ $$ y_1(x)=1-\frac{n(n+1)}{2!}x^2+\frac{(n-2)n(n+1)(n+3)}{4!}x^4-+\cdots$$ $$ ...
0
votes
0answers
46 views

Transpose/multiplication of 3D matrices

I have $A(p)=\begin{bmatrix}p_1 &p_2 & p_3\\ 2p_1 &2p_2^2 & 4p_3^3\\ 3p_1 &3p_1 & 10\\ \end{bmatrix}\tag 1$ $ p= {\left(\begin{array}{c}p_1\\p_2\\p_3\\p_4 ...
1
vote
1answer
58 views

System of ODE - Solution

I have a system of ODE to solve $$ A_{5 \times 5}\ddot{q}(t)_{5 \times 1}+ B_{5 \times 5}\dot{q}(t)_{5 \times 1}+ C_{5 \times 1} =0\tag 1$$ Given Data $A,B,C$ are constants.We know what is ...
0
votes
0answers
22 views

Matrix Algebra - Linear dependency

We have a given equation $ \frac{\mathrm{d}R(t) }{\mathrm{d} t}=R(t) \{(1-t)U_0+t U_1\}\tag 1$, all variables except scalar variable 't' has dimension $3 \times 3$. Given data $R(t)$ is ...
0
votes
1answer
34 views

Matrix Solution

I have matrix integral equation of the following form ${f^{'}(x)}_{1 \times 1}A_{3\times 3}=P_{3\times3} (1-x)+Q_{3 \times 3}x \tag 1$ . All dimensions are indicated in equation itself. " ' " ...
38
votes
15answers
6k views

Why learn to solve differential equations when computers can do it?

I'm getting started learning engineering math. I'm really interested in physics especially quantum mechanics, and I'm coming from a strong CS background. One question is haunting me. Why do I need ...
0
votes
1answer
64 views

Matrix-valued differential equation $A'(t)=A(t)B(t)$

How to solve matrix-valued differential equations of type $$A'(t)=A(t)B(t) \tag 1$$ All the given functions are square matrices of dimension $3$ and only $A(t)$ is invertible (not $B(t)$ or ...
1
vote
0answers
65 views

Matrix exponent form

We have an equation of matrix exponent $ Ae^{Ax}R-e^{Ax}R (P_1 +P_2 x) = Y \tag1$ Given condition $A,R,P_1,P_2,Y$ are constant $3 \times 3 $ matrices. R is invertible,orthonormal,determinent ...
1
vote
1answer
27 views

Diff. Eq. Example with Matrices

I'm currently working on a side project of mine that deals with $\sin(A)$ and $\cos(B)$, where $A,B\in\mathbb{C}^{nxn}$. I'm trying to find some interesting (or non-interesting) examples where one ...
1
vote
0answers
55 views

Converting a series to a recursive expression

Let $e_i$ be a unit vector with one 1 in the $i$-th element. Is the following expression has a recursive presentation? $$y = \sum_{k=0}^{\infty} {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where ...
3
votes
3answers
86 views

Conversion of rotation matrix to quaternion

We use unit length Quaternion to represent rotations. Following is a general rotation matrix obtained ${\begin{bmatrix}m_{00} & m_{01}&m_{02} \\ m_{10} & m_{11}&m_{12}\\ m_{20} & ...
1
vote
1answer
38 views

Finite-Dimensional Subspaces Invariant under Differentiation

Let $X$ be the linear space of complex continuously-differentiable functions on $\mathbb{R}$. If $M$ is a non-trivial finite-dimensional subspace of $X$ which is invariant under differentiation, does ...
0
votes
1answer
26 views

The state transition matrix formula

Let $\epsilon$ be real parameter and $\Phi(t,t_0)$ be state transition matrix. How we can prove following equation: $$ \Phi(t+\epsilon,t_0)=\Phi(t,t_0)+\epsilon\frac{d}{dt}\Phi(t,t_0)+O(\epsilon^2)\\ ...
0
votes
0answers
42 views

ODE with multiple simple conditions $f'(x)=f(x)(Ax+D ) $

I have an ODE to solve . The main issue is,in addition to solving it I have to keep some conditions too in the solution of f(x).. I am bit confused regarding how to deal with it. Equation is given ...
0
votes
1answer
53 views

First-order linear differential equation for matrix valued functions of size $3\times 3$

I have two matries given by (M' means derivative w.r.t x) $ M=\left( \begin{array}{ccc} f_1(x) & f_2(x) & f_3(x) \\ f_4(x) & f_5(x)& f_6(x) \\ f_7(x) & f_8(x) & ...
0
votes
0answers
25 views

Solving a simple Recurrence in summation form(very special case)

I have a bit confusing recursion form $\sum_{n=2}^{\infty}\{f(n)\frac{n}{n-1}\}=C, \tag 1$ $f(0)=b,f(1)= a,f(2)=c$ and $C$ are constants. Could you help me to solve this recursion or help me to ...
3
votes
2answers
109 views

Solving a particular system of differential equations

The problem I'm trying to solve is this: $X'(t) \in \mathbb{R}^3 \,, \, \omega = (\omega_1,\omega_2,\omega_3) $ Find the general solution for $$X'(t) = \omega \times X(t)$$ After doing the cross ...
0
votes
2answers
39 views

The set consisting of all solutions of a homogeneous linear differential equation of order $n$ is a vector space.

The set $S$ consisting of all solutions of a homogeneous linear differential equation of order $n$ is a vector space.
0
votes
0answers
28 views

Finding alternating series for Power series

Given data and conditions I have a power series, $PS(x) = \sum_{n=0}^\infty R_nx^n$. I have a infinite GP,something like G(x) = $\sum_{k=0}^\infty ax^k = \frac{a}{1-x} $ . Never take G(x),such ...
2
votes
0answers
32 views

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...