Tagged Questions

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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 ...
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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.
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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 $$x = Ee^{rt}$$ ...
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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 ...
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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 ...
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Show the stable age structure

Considering the population process described by where $Î³$ 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 ...
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General solution and finding eigenvalues

Find the general solution: $\hat y_1'= y_2$ $\hat y_2' = 3y_1 +2y_3$ $\hat y_3' = -y_2$ If I put it in matrix form and get the eigenvalues I got ...
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Sturm-Liouville problem: eigenvalues

I have a Sturm-Liouville problem $$y'' + \lambda^2 y = 0, \\ y'(0) + \alpha_1 y(0) = 0, \\ y'(L) + \alpha_2 y(L) = 0,$$ where $\alpha_1 \alpha_2 \neq 0$. I found that eigenvalues are ...
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A differential equation of Buckling Rod.

I tried to solve a differential equation, but unfortunately got stuck at some point. The problem is to solve the differentail equation of hard clamped on both ends rod. And the force compresses the ...
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Eigenvalue in Sturm-Liouville problem: why isn't this one valid?

I am trying to solve the following S-L problem: $\Phi_{xx}-2\Phi_x+\lambda\Phi=0$ $\Phi_x(-1)=\Phi_x(1)=0$ where $\Phi_{xx}$ is the second derivative of $\Phi$. Professor's solution is ...
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an eigenvalue problem of a differential equation

Let $L = - (\frac{d}{d \theta})^2$. Consider the eigenvalue problem $L \phi = \lambda \phi \; (a < \theta < b )$. If $\phi(a) = \phi(b) = 0$, can I derive $\lambda_n$ formula ? Here $\lambda_n$ ...
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stuck on a differential equation

let be the differential equation $x^{2} y''(x)+ y(x)(a^{2}+k^{2} _{n})=0$ the boundary conditions are $\int_{0}^{\infty}dx |y(x)|^{2} < \infty$ and $y(0)$ must be finite (regular solutions ...
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Differential equations: Connection between repeated roots of characteristic equation and generalized eigenvectors

My question is about homogeneous linear equations with constant coefficients: $ay''+by'+cy=0$. When you solve this equation via a characteristic equation (see e.g. here) problems arise when you ...
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Interpreting numerically-estimated eigenvalues of a Jacobian matrix

For small systems of differential equations, it is easy to solve for the eigenvalues of the Jacobian. From there, it is then easy to interpret what those eigenvalues mean, and to see how changing ...
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Suppose I have a first order differential operator in matrix format:- $$Dx = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 \cdots \\ -1 & 0 & ... 3answers 797 views Finding eigenvalues by inspection? I need to solve the following problem, In this problem, the eigenvalues of the coefficient matrix can be found by inspection and factoring. Apply the eigenvalue method to find a general ... 1answer 204 views Boundary value problem and eigenvalues This boundary value problem has (countably) infinitely many eigenvalues \mu_n\in \mathbb R:$$y^{(4)}-\mu y=0$$with boundary conditons: y(0)=y(1)=y'(0)=y'(1)=0 What happens to the ... 0answers 47 views Further simplifiable? I have an equation (xy')'+kxy=xf(x) where k is not an eigenvalue; y(x) and f(x) are subjected to boundary conditions (i) bounded as x\to 0; (ii) y(1)=0=f(1) I want to get the ... 0answers 235 views Differential equation, eigenvalues and eigenfunctions How does one find all the permissible values of b for -{d\over dx}(-e^{ax}y')-ae^{ax}y=be^{ax}y with boundary conditions y(0)=y(1)=0? I assume we have a discrete set of \{b_n\} where they ... 0answers 59 views Orderable eigenvalues I want to show that there are countably infinite number of eigenvalues (that can put in ascending order-- with a minimum value) to the 2nd order ODE x''+\lambda x = 0 subjected to boundary ... 1answer 122 views Eigenvalues of diff-system(can't understand) In this paper the authors have the dynamical system$$\begin{align} T_f \dot{y}_f & = -y_f + (1-\alpha(v))\varphi(z,d) \\ T_r \dot{y}_r & = -y_r + \alpha(v) \varphi(z,d) \\ \dot{z} ...
Let $\mathcal{E}$ be an ellipse in the $\mathbb{R}^2$ plane with center in $o=(0,0)$, given focal distance $c\geq 0$ and given area $A>0$. It is a fact that the eigenvalue problem for the Laplace ...