0
votes
0answers
19 views

What are the upper bound and stability conditions for the following simple linear system

Consider the following linear system $$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1) $$ where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...
2
votes
1answer
43 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 A ...
2
votes
1answer
8 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
25 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
54 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
38 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
55 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
21 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
29 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. " ' " ...
37
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
59 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
62 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
25 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
53 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 ...
2
votes
3answers
70 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
36 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
18 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
45 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
24 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
94 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
36 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
27 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
31 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} = ...
0
votes
1answer
23 views

What is the Jacobian of the following function

Consider a function F: $R^n \to R^n$ defined by $$f(u) = A*u*(n+1)+\lambda *B$$ Where A is a tridiagonal n-by-n matrix with -2 on the main diagonal and 1 on the off diagonals. B = $\begin{pmatrix} { ...
0
votes
0answers
17 views

Matrix multiplier for ODE

I have matrix C with dimensions $3 \times 3 $ and it is skew symmetric too C is given by $C(0,0)=0,C(1,1)=0,C(2,2)=0 \tag 1$ $C(1,0)= sc_0+ px (c_1-c_0),C(0,1)=-C(1,0) \tag 2 $ $C(0,2)= ...
0
votes
1answer
23 views

Question related to vector space of solutions of a differential equations system.

I have some doubts regarding the proof of the statement: The set of solutions of the system $$X'=A(t)X$$ where $A(t):I \subset \mathbb R \to \mathbb R^{n \times n}$ is continuous, forms an ...
2
votes
1answer
60 views

Proving that a zero Wronskian implies linear dependence without Abel's theorem

Claim: If the Wronskian of $x_1,\dots x_n$ is zero and $L(x_i)=0, \quad i=1,\dots,n$, where $L$ is a linear differential operator or order $n$, with continuous coefficients, then $\{x_1,\dots,x_n\}$ ...
0
votes
0answers
34 views

Finding the largest eigenvalue of a sparse matrix

I would like to find the largest eigenvalue of a sparse matrix by hand- this is part of analyzing a mathematical model for infectious diseases. The nonzero entries are very complicated - hence Maple ...
2
votes
2answers
68 views

How to determine the eigenvectors for this matrix

I have the matrix $$\left( \begin{array}{ccc} -\alpha & \beta \\ \beta/K & -\alpha/K \end{array} \right)$$ for which the eigenvalues are ...
2
votes
3answers
40 views

Linearly Independency of functions

Show if the functions are linearly independent $x(t)=3$, $y(t)=3\sin^2t$, $z(t)= 4\cos^2t$ How can i show this?
1
vote
1answer
23 views

Solving $x' = Ax$ for real $x$ where $A$ is a matrix with complex eigen values

I have the following linear differential equation system: $$x' = A x$$ where $$ A = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 3 & 1 & -2 \\ 2 & 2 & 1 \end{array} \right) $$ I ...
0
votes
1answer
20 views

Stability of system of nonlinear differential equations

In order to find the stability of a nonlinear system of differential equations (in the real plane) we need to show that the eigenvalues of the linearized system are all negative. Can someone explain ...
1
vote
0answers
25 views

Linear homogeneous ODE system of first order

Good afternoon. I recently encountered the following problem to which I couldn't find a solution anywhere so far: Given $A:D\to\mathbb C^{2\times 2}$, $D\subset\mathbb C$ open, with holomorphic ...
2
votes
0answers
145 views

Solving an infinite non autonomous system of differential equations.

For all $\lambda\in\mathbb{R}$, let $J(\lambda)$ be the infinite matrix where $(J(\lambda))_{nn}=\lambda$, $(J(\lambda))_{n,n+1}=1$ for all $n\in\mathbb{N}$, and all other entries are $0$. This matrix ...
0
votes
1answer
44 views

General solution of a system of linear differential equations with multiple generalized eigenvectors

I am looking for general solutions for the linear sODE's $$\textbf{x}'(t) = A\textbf{x}(t)$$ with $t \geq 0$ and $A \in \mathbb{R}^{n \times n}$ Let focus on just real eigenvalues and eigenvectors. ...
0
votes
2answers
65 views

Basis of solution sets

I know that the collection of all solutions to $\sum_{i=0}^nA_iy^{(i)}(t)=0$ form a vector space. But in which way can one find out its basis? Of course I already learned what the basis is. But the ...
2
votes
0answers
47 views

How to solve this system of inhomogeneous differential equations

In some past exam papers for the Maths course that I attend,I found this example and I would really appreciate if someone looked at my solution. It goes like this: Find general solution to $$ y_1' = ...
1
vote
0answers
13 views

Finding normal components of a vector

If \begin{align} \notag A_{1}=c_{1}y_{1}^{2}y_{2}^{2}u(u+2t)(u+y_{2}^{2})\frac{\partial}{\partial x_{2}} \end{align} and \begin{align} \notag A=-c_{11}\frac{\partial}{\partial ...
3
votes
1answer
43 views

Similarity transformation of a linear operator

I've seen in some books that given a differential operator $$\frac{d}{dx}$$ under a similarity transformation we get $$\frac{d}{dx}\rightarrow ...
2
votes
2answers
76 views

How to solve this system of equations that appears in a ODE exercise?

I am trying to solve this equation, we know $A, B, Q,\phi\in\mathbb{R}$. \begin{eqnarray} T''(x) &=& \phi (T(x)-Q) \\ T(0)&=& A\\ T(b)&=&B \end{eqnarray} So the ...
2
votes
1answer
88 views

Solving the matrix differential equation $v'(z) = A e^z + B v(z), v'(0) = 0)$

Take the system of ODEs in terms of the scalar $z$, $$v'(z) = A e^z + B\cdot v(z)$$ With the initial condition, $$v'(0) = \mathbf{0}$$ For some $B$ an $N \times N$ matix, and $A$ a $1\times N$ ...
1
vote
0answers
16 views

Is there any relation between positive definite operator and an operator that satisfies maximum principle?

Suppose $L$ is a self adjoint differential operator which satisfies maximum principle. Maximum principle: Assume that $u(x)$ satisfies $u(0)\geq 0$ and $u(1)\geq 0$. Now $L$ is said to satisfy ...
0
votes
0answers
38 views

Problem with commutator relations

part a) is fine. part b) is not. A commutator is defined as, for operators $A$ and $B$, $[A,B]=AB-BA$. [SOLVED]I get that $H(\lambda)=e^{-\lambda D}Ce^{\lambda D}$, $H'(\lambda)=-De^{-\lambda ...
1
vote
1answer
37 views

Solving Matrix Equations with Extra Vector Term

How does one solve an equation of the form $$\vec{x} = A \vec{a} + \vec{b} $$? Naturally if one wants to solve $$\vec{x} = A \vec{a}$$, we compute eigenvalues and eigenvectors of $A$, but what do we ...
0
votes
1answer
48 views

Null space in ordinary differential equations

I've been watching some videos on linear algebra on OCW, and alternating between thinking I "get" nullspace, and not knowing what's going on at all. Just now however, I did a problem set with the ...
0
votes
1answer
89 views

Verify that the given vector satisfies the given differential equation

$$\vec x' = \begin{bmatrix}3 & -2 \\ 2 & -2\end{bmatrix} \vec x; \qquad \vec x = \begin{bmatrix}2 \\ 1\end{bmatrix} e^{2t}.$$ So I was wondering how can I verify the vector satisfies the ...
2
votes
1answer
46 views

Find $e^{AT}$ where $A$ is a Matrix that is given

How to find the value of $e^{At}$ where $A$ is the matrix $A =\begin{bmatrix} 4 & 3 \\ 2 & -1 \end{bmatrix}$
1
vote
1answer
38 views

Why is the fundamental matrix of a linear system of ODEs always invertible?

Why does $\phi^{-1}(0)$ exist, where $\phi(t)$ is the fundamental matrix of the system $\dot{x}=A(t)x$, $x \in \mathbb{R}^n$? I am not able to figure this out.
2
votes
0answers
33 views

Interior Products

Over on the Wiki page for interior products: http://en.wikipedia.org/wiki/Interior_product There is a line that says $\iota_X \alpha = \alpha(X) = \langle \alpha,X \rangle$ where $\alpha$ is a ...