# Tagged Questions

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### 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: ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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, ...
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### 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 $A$ ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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) ...
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### 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 ...
A linear system and its general solution. $dx/dt$ = $6x - 2y$ $dy/dt$ = $4x + 2y$ It has a general solution of this: $$\begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = A\begin{bmatrix} cos(2t) \\ ... 2answers 485 views ### Using Eigenvalues and Eigenvectors, Find the general solution of the following coupled differential equations. x'=x+y and y'=-x+3y. Consider the matrix A=\begin{bmatrix} 1 & 1 \\ -1 & 3 \end{bmatrix} I found the eigenvalue \lambda=2 with multiplicity 2. However, the general solution I found degrees with the answer ... 1answer 390 views ### Eigenvalues and Eigenfunctions of a singular Sturm-Liouville operator using Bessel functions I’m trying to find the eigenvalues and eigenvectors of the Singular Sturm-Liouville operator:$$Lu=xu''+u'u(1)=0u(0) \text{ is finite}0 < x < 1My approach to solving ... 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 ... 1answer 73 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} ... 2answers 79 views ### Solving a linear system with complex eigenvalues I have the system: $$x' = \begin{pmatrix}5&10\\-1&-1\end{pmatrix}x$$ The corresponding characteristic equation is: \lambda^2-4\lambda+5 \\ \implies ... 1answer 130 views ### two dimensional linear differential equation with 1 eigenvector I have the following linear differential equation: $$x' = \begin{pmatrix}3&-4\\1&-1\end{pmatrix}x$$ The corresponding characteristic equation is: ... 1answer 406 views ### Finding eigenvalues and eigenfunctions for a BVP Find the eigenvalues and eigenfunctions fory'' + \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 ...
I have the following differential equation which is motivated by the dynamics of a mass on a spring: $$my'' - ky = 0$$ I split this into a system of equations by letting ...