0
votes
0answers
21 views

Why do we want that the determinant of the coefficients is $0$?

Eigenvalue problem with periodic boundary conditions-complete Fourier series $$y''+\lambda y=0, 0 \leq x \leq L$$ $$(*): \begin{cases} y(0)=y(L)\\[4pt] y'(0)=y'(L) \end{cases}$$ $$$$ It's a ...
2
votes
0answers
44 views

Question about an eigenvalue problem

I have a question... How can I show that the eigenvalue problem $$y''+λy=0$$ $$y(0)=0,$$ $$ y'(0)=\frac{y'(1)}{2}$$ is NOT a Sturm-Liouville problem?
1
vote
0answers
25 views

3-Species Population Model

I am trying to solve a 3-species predator-prey system in matlab. Here is the equation: $$\frac{d}{dt} \begin{bmatrix} N_1 \\ N_2 \\ N_3 \\ \end{bmatrix} = \begin{bmatrix} N_1 & 0 & 0 \\ 0 ...
0
votes
1answer
18 views

How would you compute eigenvectors from this linear system?

I am stuck on a problem and I do not know how to obtain the eigenvectors: $\frac{dY}{dt}=\bigl(\begin{smallmatrix} -2&0\\ -3&1 \end{smallmatrix} \bigr)Y$ Work: I obtained the eigenvalues ...
0
votes
1answer
26 views

Proof of Lyapunov Stability for Constant Matrix System

I am trying to find the necessary and sufficient conditions for the point of equilibrium x=0 of $x'=Ax$ to be Lyapunov stable, where A is constant matrix. The book I'm using briefly touches on this, ...
4
votes
2answers
70 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 $A$ ...
1
vote
2answers
84 views

How to solve this differential equation system?

The following system is given: $$ \dot{x} = y + z \\ \dot{y} = x + z \\ \dot{z} = x + y $$ The first thing I did was to find out the eigenvalues. I found out, that -1 is a doubled and 2 a single ...
0
votes
1answer
69 views

Systems of linear differential equations - eigenvectors

Solve the following system of equations $ \begin{cases} x_1^{'}(t)=x_1(t)+3x_2(t) \\ x_2^{'}(t)=3x_1(t)-2x_2(t)-x_3(t) \\ x_3^{'}=-x_2(t)+x_3(t)\end{cases} $. First, I create the column vectors ...
2
votes
1answer
61 views

Use of Routh-Hurwitz if you have the eigenvalues?

This is for self-study of N-dimensional system of linear homogeneous ordinary differential equations of the form: $$ \mathbf{\dot{x}}=A\mathbf{x} $$ where A is the coefficient matrix of the system. ...
0
votes
2answers
33 views

Prove that trajectory that starts in span of eigenvector will remain there

Assume we have a 2-d system of homogeneous ordinary differential equations:$$ \dot{\mathbf{x}}=\left[ \begin{array}{ c c } a & b \\ c & d \end{array} ...
1
vote
1answer
23 views

Jordan canonical forms and deficiency indices

I'm solving a homework question that asks me to do the following: "List the five upper Jordan canonical forms for a $4\times 4$ matrix $A$ with a real eigenvalue $\lambda$ of multiplicity $4$ and ...
2
votes
1answer
168 views

How to match eigenvalues with directional field graphs?

How they were able to match eigenvalues with the graphs?
4
votes
0answers
59 views

Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
0
votes
0answers
36 views

Convert an eigenvalue equation to ODE/s

For example define: $K=-i\frac{d}{dx}$ (non-discrete spectrum), so: $$Kf(x)=-i\frac{df}{dx}=kf(x)$$ Define $g(x,k)=kf(x)$, so: $$\frac{-i}{k}\frac{\partial{g}}{\partial{x}}=g(x,k)$$ ...
4
votes
2answers
54 views

Nonhomogeneous Linear ODE

$$ x' =\left(\begin{array}{rr}4 & 8 \\ -2 & -4\end{array}\right)x + \left(\begin{array}{rr}t^{-3} \\ -t^{-2}\end{array}\right), t>0 $$ To find the general solution of the given system ...
0
votes
1answer
24 views

Repeated Eigenvalues in Systems of ODEs

Question is to find the general solution of the given system of equations below. $$ x' =\left(\begin{array}{rr}\frac{-3}{2} & \frac{-1}{4} \\ 1 & \frac{-1}{2}\end{array}\right)x $$ My ...
0
votes
1answer
59 views

Fundamental matrix for a given system of equation

Question is to find the fundamental matrix(F(t)) satisfying F(0)=I for the given system of equation below. $$ x' =\left(\begin{array}{rr}2 & 3 \\ -1 & -2\end{array}\right)x $$ My solution ...
1
vote
2answers
87 views

IVP with challenging numbers. I've seen it evaluated by non trivial manipulations. Can someone complete it step by step with explanations?

Solve the IVP: $X' = AX+f(t)$ $$\begin{align*} A&= \begin{bmatrix}6/7 & -15/14\\-5/7 & 37/14\end{bmatrix} \\ X(0)&= \begin{bmatrix}4\\-1\end{bmatrix} \\ f(t)&= ...
0
votes
2answers
102 views

Concerning the general solutions to linear ODEs Y'=AY when A has multiple eigenvalues

Given linear ODES Y'=AY, where Y is a column vector, A is a 6*6 square matrix. Clearly A has 6 eigenvalues, namely r1, r2, r3, r4, r5, r6. Herein we assume r5=r2, r6=r3.That is, r2 and r3 are two ...
0
votes
0answers
41 views

Regular Sturm-Liouville Boundary Value Problem

Let $L[y]:=y''''$. Let the domain of $L$ be the set of functions that have four continuous derivatives on $[0,π]$ and satisfy $y(0)=y'(0)=0$ and $y(π)=y'(π)=0$ a) Show that $L$ is self adjoint b) ...
1
vote
1answer
67 views

Repeated Eigenvalues Initial Value Problem

If someone could help me step by step in solving this initial value problem. There's a lot I'm confused about since the solution is supposed to be expressed in the form $x_1(t) =$ and $x_2(t) =$: $$ ...
2
votes
1answer
106 views

Finding eigenvalues and eigenfunctions.

Find eigenvalue and eigenfunction of $x^2y''-xy'+(\lambda+1)y=0$ with $y(1)=0=y(e)$. I have found the equation to be Euler-cauchy equation ,so i try y=x^n, which yields: $n=1+\lambda i$ thus ...
0
votes
1answer
69 views

Duplicate zero eigenvalue

What happens when you have two zero eigenvalues (duplicate zeroes) in a 2x2 system of linear differential equations? For example, ...
1
vote
1answer
83 views

Markov Chain Solution Eigenvalue

I am having trouble understanding how to solve for the state vector at time $t$ for a markov chain using matrix algebra. I have the following Markov Transition Intensity Matrix, for the states A, N, ...
1
vote
1answer
93 views

Eigenfunction expansion

Use the appropriate engenfunction expansion to represent the best solution. $$u''=f(x), u'(0)=\alpha, u'(1)=\beta$$ I use the function $$\phi''+\lambda\phi=0$$ to get the eigenfunction is ...
0
votes
1answer
65 views

Eigenvalue problem?

"Solve the eigenvalue problem or show that it has no solution: $y'' + 2y = x$ for $y(0) = y(\pi) = 0$" I have managed to find a solution to the boundary value problem by finding the complementary ...
1
vote
0answers
41 views

Given $\mathbf{A}$ stable (all negative eigenvalues), produce a bound on $\|\mathbf{B}\|$ such that…

Given a system: $\dot{\mathbf{x}}=\left(\mathbf{A}+\mathbf{B}\right)\mathbf{x}$ Can you bound $\|\mathbf{B}\|$ s.t. the origin of $\mathbf{x}$ is exponentially stable using a Lyapunov function? ...
1
vote
0answers
76 views

Sturm-Liouville Eigenvalues

Consider Sturm-Liouville endpoint problems of the form $y''+\lambda y=0$ with the usual endpoint conditions. $c_1y(a)+c_2y'(a)=0$, $d_1y(b)+d_2y'(b)=0$. Here $(c_1,c_2) \neq \vec{0}$ and $(d_1,d_2) ...
2
votes
1answer
99 views

Compute $e^{tA}$

When I do my homework (stability theory), I must use the knowledge to the matrix. But I don't remember it :(. Here's my problem: For the system of equations: $$\begin{cases} & \text{ } ...
0
votes
3answers
69 views

another generalized eigenvector question

I have $$ A = \left( \begin{array}{ccc} -4 & 9 & -4 \\ 0 & 0 & 0 \\ 6 & -13 & 6 \end{array} \right) $$ whose eigenvalues are $\{0,0,2\}$. For $\lambda=2$, I have ...
2
votes
1answer
670 views

How to sketch the phase portrait near the critical point at the origin.

A linear system and its general solution. $dx/dt$ = $6x - 2y$ $dy/dt$ = $4x + 2y$ It has a general solution of this: $$\begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = A\begin{bmatrix} cos(2t) \\ ...
2
votes
2answers
290 views

Using Eigenvalues and Eigenvectors, Find the general solution of the following coupled differential equations. x'=x+y and y'=-x+3y.

Consider the matrix $A=\begin{bmatrix} 1 & 1 \\ -1 & 3 \end{bmatrix}$ I found the eigenvalue $\lambda=2$ with multiplicity $2$. However, the general solution I found degrees with the answer ...
1
vote
1answer
355 views

Eigenvalues and Eigenfunctions of a singular Sturm-Liouville operator using Bessel functions

I’m trying to find the eigenvalues and eigenvectors of the Singular Sturm-Liouville operator: $$Lu=xu''+u'$$ $$u(1)=0$$ $$u(0) \text{ is finite}$$ $$0 < x < 1$$ My approach to solving ...
1
vote
1answer
45 views

Solving Linear ODE using matrices

What I don't understand here is where or how the operator for this solution is formed. Shouldn't the values of the operator be A=(1,0,0,1)? (in the form a11, a12, a13, a14 respectively). Any help ...
2
votes
1answer
60 views

Finding a Hopf Bifucation with eigenvalues

I am trying to show that the following 2D system has a Hopf bifurcation at $\lambda=0$: \begin{align} ...
1
vote
2answers
74 views

Solving a linear system with complex eigenvalues

I have the system: \begin{equation} x' = \begin{pmatrix}5&10\\-1&-1\end{pmatrix}x \end{equation} The corresponding characteristic equation is: \begin{equation} \lambda^2-4\lambda+5 \\ \implies ...
1
vote
1answer
113 views

two dimensional linear differential equation with $1$ eigenvector

I have the following linear differential equation: \begin{equation} x' = \begin{pmatrix}3&-4\\1&-1\end{pmatrix}x \end{equation} The corresponding characteristic equation is: \begin{equation} ...
0
votes
1answer
355 views

Finding eigenvalues and eigenfunctions for a BVP

Find the eigenvalues and eigenfunctions for $$y'' + \lambda y = 0, y(0) = 0, y'(\pi/2) = 0$$ According to my book we must check 3 cases: $\lambda < 0$, $\lambda = 0$, $\lambda > 0$. I started ...
2
votes
1answer
163 views

Differential equation of a mass on a spring

I have the following differential equation which is motivated by the dynamics of a mass on a spring: \begin{equation} my'' - ky = 0 \end{equation} I split this into a system of equations by letting ...
2
votes
2answers
78 views

Spectrum of Lyapunov exponents of a linear system

Question: How to show that the eigenvalues of matrices $\mathbf{A}$ and $ \mathbf{L} = \log \lim_{t \to \infty} \left((e^{\mathbf{A}t}e^{\mathbf{A^T}t})^{\frac{1}{2t}}\right) $ have equal real parts? ...
1
vote
0answers
34 views

First eigenvalue of the given linear operator

I have the following question: Let us denote $H_2^N: = \{u\in (H^2(0,1))^2: u'(0) = u'(1) = 0\}$. Let an operator $L:H_2^N \to (L^2(0,1))^2$ be given by $Lu = -Du'' + Cu$, where $D$ is a positive ...
1
vote
1answer
217 views

Considering the linear system Y'=AY

What would be an equation that I can use when I compute the eigenpairs for the coefficient matrix A.
1
vote
1answer
67 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 $$ x = Ee^{rt}$$ ...
2
votes
1answer
71 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
42 views

Are the eigenvalues limited?

While studying elliptic operators, I encountered the following problem, which I'm having problems to prove or give a counter-example: Let $\Omega$ be an open subset of $\mathbb{R}^m$, and suppose ...
0
votes
1answer
110 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
0answers
79 views

Solving a Sturm-Liouville differential equation variationally

This is a problem from Haim Brezis' functional analysis book (Exercise 8.41). I solved parts of it, but am stuck on some parts/want confirmation on the method. The problem is as follows: Let $q ...
1
vote
1answer
74 views

Boundary-value problem in differential equations

Consider the problem: $$u^{(4)} + \lambda u = 0, \ \ \ 0<x<\pi; \ \ \ u(0) = u(\pi) = u''(0) = u''(\pi) =0$$ Find the eigenvalues. How should one proceed about this problem? I am complete ...
2
votes
3answers
216 views

Spectrum of eigenvalues and eigenfunctions

Our O.D.Es professor had the "amazing" idea of heavily introducing advanced linear algebra material (which is not an official prerequisite for the course) along with boundary value problems. Not being ...
0
votes
2answers
140 views

Eigenvalue of some Sturm–Liouville problem

I have one simple question. How I suppose to show that $\lambda =0$ is an eigenvalue of some problem. Does it mean that I must have non-trivial solution for $\lambda=0 $? Thanks! UPD:I mean by ...