1
vote
1answer
34 views

defenite integral involve bessel function

I have an integral which involves Bessel function as follows: $I=\int_{r=0}^a \int_{\theta=0}^{2\pi}(e^{-jkr\cos(\theta-\phi)}d\theta)rdr$ I have tried with $e^{-jkr\cos(\theta-\phi)}=\sum ...
2
votes
1answer
57 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
28 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
60 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
39 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
22 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
19 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
86 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
35 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
42 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
35 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
75 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
85 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
13 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 ...
-8
votes
1answer
60 views

Finding general solution, show work [closed]

Find the general solution to $y'-3xy=xy^{1/3}$. Need answer and work for a final review.
0
votes
0answers
33 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
42 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
43 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
39 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
28 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.
1
vote
0answers
24 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 ...
0
votes
0answers
24 views

5 Parameter Affine Transformation

I am working on computing affine transformation using Gradient Ascent Method, so the Inverse compositional algorithm. However, I am stuck in one probably simple step but I fail to understand them. ...
2
votes
2answers
36 views

Solving bernoulli differential equation

How to solve $$t \frac{dy}{dt} + y = t^4 y^3$$ First I divided by $t$ to get $$\frac{dy}{dt} + \frac{y}{t} = t^3 y^3$$ Then I multiplied through by $y^{-3}$ to get $$y^{-3} \frac{dy}{dt} + ...
1
vote
2answers
79 views

Finding a particular solution to the non-homogenous system

I have the following problem $\vec{x}^{'}(t)=\begin{pmatrix} 2 & -5\\1 & -2 \end{pmatrix}\vec{x} + \begin{pmatrix} \csc t\\ \sec t \end{pmatrix}$ Step 1) Find the Eigenvalues ...
0
votes
2answers
28 views

Inverse of matrix with varying parameters

Ok so I need some sort of verification on this. I have run into this matrix $\begin{pmatrix} e^t&3e^{-t}\\e^t&e^{-t} \end{pmatrix}$ and I need to find the inverse of this matrix. The book ...
2
votes
1answer
49 views

Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ...
2
votes
1answer
34 views

Suppose $\mu$ is not an eigenvalue of A. Show that the equation $x'= Ax + e^{\mu t}b$.

Suppose $\mu$ is not an eigenvalue of $A$. Show that the equation $x'= Ax + e^{\mu t}b$ has a solution of the form $\varphi(t) = ve^{\mu t}$.
2
votes
1answer
31 views

Solution in common for two differential equations

Consider: $E1: y''-4y'+4y=0$ Solution: $y(x)=c_1 e^{2x}+c_2 x e^{2x} $ $E2: y''-2ay'+(a^2-1)y=0$ Solution: $y(x)=c_1 e^{(a+1)x}+c_2 e^{(a-1)x} $ For what values of $a$, $E1$ and $E2$ have ...
1
vote
1answer
38 views

Problem with checking whether $x(t)$ can be a solution of any system of first order homogeneous ODE

I need to find out whether $$x(t) = (3e^t + e^{-t}, e^{2t})$$ can be a solution of the system $$\dot{x} = A x\quad \quad (1)$$, where $A$ is a $2x2$ matrix. I'm not sure of my solution, which is the ...
2
votes
2answers
66 views

Systems of Linear Differential Equations - Is this Correct?

I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem: (Note: $'$ denotes prime) ...
3
votes
2answers
46 views

System of differential equations using substitution

Exact problem statement Solve the system $\left\{\begin{matrix} x_{1}'(t)=3x_{1}(t)-2x_{2}(t)+e^{2t},x_{1}(0)=a & \\ x_{2}'(t)=4x_{1}(t)-3x_{2}(t),x_{2}(0)=b & \end{matrix}\right.$ by using ...
3
votes
2answers
95 views

Using the Jordan form Complex

Let $C$ be a complex $n \times n$ matrix with $\det C \neq 0$. Prove that there is a complex matrix $B$ such that $C = e^B$ Hint: use the Jordan form matrices for comlexas
3
votes
1answer
54 views

Systems of Linear Differential Equations - population models

I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem: (Note: $'$ denotes prime) ...
0
votes
0answers
32 views

Ordinary Differential Equation and linear algebra

Consider the ODE $$-u_{k+1} + 2u_{k} - u_{k-1} = \frac{k}{(n+1)^{3}}$$ With the conditions $u_{0}=0$ and $u_{n+1}=0$, $k=1,...,n$. The points $P_{k} = \left( \frac{k}{n+1}, u_{k}\right)$, $k= ...
1
vote
1answer
89 views

differential equations, diagonalizable matrix

I have a question of differential equations of the form. $\textbf{x}'(t)=A*\textbf{x(t)}$, where x is an n-dimensional matrix, and A is an n*n real matrix. I have learned to solve this if a is ...
0
votes
0answers
21 views

Linear gradient equation in the plane

Observe the following equation: $V(y)=\frac{a}{2}y_1^2+\frac{b}{2}y_2^2+c y_1y_2$ a) Find the matrix $A$ which you find on the right hand side of the equation. b) Calculate the Trace, determinant, ...
1
vote
0answers
21 views

joint density random variables with a set of equations

There are $n$ equations: $f_i(x_1,x_2,...,x_n,e_i)=0$, $i$ from $1$ to $n$, where $e_i$ are independent random variables whose expectations are all $0$. $x_i$ are random variables. Suppose the map $e ...
1
vote
1answer
23 views

Diff EQ. Problem (Eigenvector issue)

Find the general solution of $\textbf{x}^{'}=\begin{pmatrix} -1&-4\\1&-1\end{pmatrix}\textbf{x}$. The eigenvalues I found are $-1 \pm 2i $ and I chose $-1-2i$ to be my eigenvector and found ...
1
vote
1answer
29 views

Finding the General Solution to the system of equation

Find the General solution of $\textbf{x}^{'}=\begin{pmatrix} 2&2+i\\-1&-1-i\\ \end{pmatrix}\textbf{x}$ I started out by finding the eigenvalues. ...
4
votes
2answers
81 views

Show each eigenvalue of a companion matrix has geometric multiplicity $=1$.

Given the differential equation $$x^{(n)}(t)+c_{n-1}x^{(n-1)}(t) + \dotsb + c_1x'(t) + c_0=0,$$ we can form a vector $\xi = (x, x', \dotsc, x^{(n-1)})$, and then we have $$\xi'(t) = A\xi,$$ where $A$ ...
0
votes
1answer
33 views

First-order linear differential equation

I have this question, and the working out below is as far as I can get: $$ x \frac{dy}{dx} - y = y^2 \\ p(x) = -\frac{1}{x} \\ q(x) = \frac{y^2}{x} \\ u(x)= e^{\int -\frac{1}{x}dx} \rightarrow ...
-1
votes
2answers
136 views

Eigenvalues of $d/dx$.

Consider $d/dx:C^\infty(\mathbf{R})\rightarrow C^\infty(\mathbf{R})$ (both as real vector spaces). I want to find its eigenvalues and corresponding eigenvectors. Every $\lambda\in\mathbf{R}$ is an ...
0
votes
1answer
38 views

How to solve this second order linear homogeneous ODE?

I'm trying to solve $\frac{d^2y}{dt^2}+\frac{2dy}{dt}+10y=0$ Initial values $t=0$, $y=1$, $\frac{dy}{dt}=0$, $h=\frac{1}{1000}$. Problem: Find $y$ when $t=1$ using the Heun Method to approximate ...
2
votes
1answer
53 views

Differential linear system limit

Suppose that $u\in \mathcal{L}(\mathbb{R}^n)$ and $$\forall x\not\in \ker u,\quad (x,u(x))<0,$$ where $(.,.)$ is a scalar product on $\mathbb{R}^n$. Then consider $$ \left\{ ...
0
votes
1answer
37 views

Determine value a in system matrix

I'm trying to solve the following problem: "Look at the image of trajectories of a linear, time-invariant system with the form: $\frac{d\textbf x}{dt}=\textbf {Ax}:$ Determine possible eigenvectors ...
1
vote
1answer
22 views

Differential equations and surjectivity of some linear operators

Let $a_0,a_1,...a_{n-1}$ be some continuous functions $[0,1]\longrightarrow \mathbb{R}$. Consider a linear operator $D:C^n[0,1]\longrightarrow C[0,1]$ which maps each $y\in C^n[0,1]$ to ...