3
votes
2answers
39 views

Eigenvalues of Differential Equation with Boundary Condition

Here is a problem from my homework assignment that I am struggling with: Consider the differential equation $\frac{d^2\phi}{dx^2}+\lambda\phi=0 $. Determine the eigenvalues $\lambda$ if $\phi$ ...
0
votes
0answers
19 views

Eigenvalue of Heun's function and its computation

It is known that the Heun's differential equation: \begin{equation} \frac{d^2 w}{dz^2} + (\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-a})\frac{dw}{dz}+\frac{\alpha \beta z -q}{z(z-1)(z-a)} ...
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 ...
0
votes
0answers
19 views

Find real solution for an inhomogene system

I have an inhomogene differential equation system $\begin{pmatrix}\dot{x}_1 \\ \dot{x}_2\end{pmatrix} = \begin{pmatrix}-1 & 3 \\ -3 & -1\end{pmatrix} \begin{pmatrix}x_1 \\ x_2\end{pmatrix} + ...
1
vote
0answers
17 views

Sturm-Liouville equation with rational coefficient

I am trying to solve a regular Sturm-Liouville type Ordinary Differential Equation (ODE) with the following form: \begin{equation} \frac{d}{dy} [q(y) \frac{dW(y)}{dy}]+p(y) W(y) = -\lambda W(y) ...
0
votes
2answers
82 views

Eigenvalues for the Sturm-Liouville boundary value problem

Please show me how to calculate the eigenvalues for the following boundary value problem: $$x''+\lambda x=0\\x(0)=0\\x(\pi)=0\\x'(\pi)=0$$ This is what I did: let $\lambda=\mu^2$ $$X(x)=A\cos\mu ...
0
votes
0answers
34 views

Show there exists a unique solution to $-u''(x)+\lambda \int^1_0 \sin(u(y))dy =f(x)$

Let $\lambda\in (-1,1)$. Show that for every $f\in C[0,1]$ there exists a unique solution $u\in C[0,1]$ to $$-u''(x)+\lambda \int^1_0 \sin(u(y))dy =f(x)$$ With $u(0)=u'(1)=0$. My work thus far: ...
4
votes
1answer
40 views

What is the purpose of studying Sturm-Louville eigenvalue problem?

After a cursory read on the SL eigenvalue problem, I did not immediately feel enlightened and failed find much usefulness except for knowing that SL generalizes a broader class of differential ...
0
votes
1answer
25 views

Equivelent solutions to second order ODEs

I have worked out the solution to an ODE using two methods, with solutions as follows: $$A e^{-5t} + B e^{3t}$$ $$\text {and}$$ $$A\begin{pmatrix} \;1 \\ {-5} \end{pmatrix} e^{-5t}+B\begin{pmatrix} 1 ...
4
votes
1answer
77 views

Interpretation of generalized eigenvector in orbits

First of all, this is my fourth question about dynamical systems in a week, sorry for that. Considering a linear bidimensional dynamical (autonomous) system, the orbits can be plotted in the phase ...
0
votes
1answer
30 views

In what cases are the eigenvalue equal to the pole points?

I have a transfer function in form of a matrix and want to determine the stability of the whole system. Now I'm wondering if I need to calculate the pole points or the eigenvalue. A friend of mine ...
2
votes
0answers
42 views

Differential Equations and Eigenvalues

I have the following system of differential equations: $$\left\{\begin{aligned} \frac {dx} {dt}=-4x+2y \\ \frac {dy} {dt}=-\frac 5 2x+2y \end{aligned} \right. $$ Which corresponds to the following ...
0
votes
1answer
53 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. ...
-3
votes
1answer
53 views

Why does it stand that $Lu=\lambda u$?

I am looking at the chapter of eigenvalue problems and I have a question.. Why does it stand that $$Lu=\lambda u$$? where $L$ is the differential operator.
0
votes
0answers
23 views

System of coupled Sturm-Liouville equations

So I have a system of two coupled Sturm-Liouville equations that can be written as follows: since $-\frac{d}{dx}(p(x)\frac{dy}{dx})+q(x)y=\lambda\omega(x)y$ we thus have p=w=1 (or the identity ...
1
vote
2answers
53 views

Euler method application: step size

Suppose we have a system of ODE's: $a' = -a - 2b$ and $b' = 2a-b$ with initial conditions $a(0)=1$ and $b(0)=-1$. How can we find the maximum value of the step size such that the norm a solution of ...
1
vote
1answer
49 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 ...
1
vote
0answers
54 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
28 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
19 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 ...
5
votes
0answers
87 views

Examples of potentials for which Schrödinger equation lacks discrete points in continuous spectrum

In Landau, Lifshitz, "Quantum Mechanics, non-relativistic theory" in $\S18$ "The fundamental properties of Schrödinger's equation" the following is said in a footnote: it must be mentioned that, ...
0
votes
1answer
36 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
92 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
95 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
84 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
114 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
39 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
50 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
176 views

How to match eigenvalues with directional field graphs?

How they were able to match eigenvalues with the graphs?
5
votes
0answers
69 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 ...
1
vote
0answers
41 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
59 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 ...
1
vote
1answer
32 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
91 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
137 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
56 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
102 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
115 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
152 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
95 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
105 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
68 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
48 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
84 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
105 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
78 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
1k 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
527 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
392 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 ...