Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.
2
votes
0answers
77 views
IVP Perturbation With Small Non-Linear Term
EDIT: Sorry to bump this without having anything extra to add, but I still cannot reconcile my solution with what was asked (in (2)). Could someone with expertise in this subject take a look? I ...
2
votes
0answers
108 views
Prove that the first positive root of the solution to the Lane-Emden equation increases steadily with $n$.
Let $\lambda$ be the first positive value for which $y=0$ where $y(x)$ satisfy the following differential equation
$$
y''+\frac{2}{x}y'+y^n=0,\qquad\text{where }n\in\mathbb{R},\ y(0)=1,\text{ and }\ ...
2
votes
0answers
98 views
System of ODEs with a degenerate (?) critical point
I am not sure the name for this is really "degenerate", but consider the following system of non-linear ODEs:
\begin{align*}
\frac{dx}{dt} & = a(1-x)z-ex, \\
\frac{dy}{dt} & = -axy, \\
...
2
votes
0answers
140 views
How to check if ode system is gradient system?
How do I check for a given (nonlinear) system of ODEs if this is a gradient system? Meaning how do I check the existence of a pseudo Riemann metric and a potential function? May be somebody could post ...
2
votes
0answers
97 views
Check my solution - Modelling of a spring with Differential Equation
I am doing some work with differential equations. I have solved the following problem but am uncertain if I'm doing it correctly. Could someone look over it for me and check if I'm doing something ...
2
votes
0answers
79 views
Solve $x^2u''+xu'-(x^2+\frac{1}{4})u=0$ using power series
I stumbled upon this question in an old exam (I'm preparing for an exam of a course about ODEs). I didn't have much difficulty solving the Legendre and Hermite equations using power series, but this ...
2
votes
0answers
59 views
$u''+\frac{4}{x+1}u'+\frac{2}{\left(x+1\right)^{2}}u=0$ variational solution
This is a concept solution scheme derived from a particular example that I have not been able to generalise sufficiently. The objective is to find a particular solution to a certain second-order ...
2
votes
0answers
46 views
Lower bound for the eigenvalue
For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined
$$
...
2
votes
0answers
118 views
Hermite functions and integral
Let
$$
h_n(x)=(-1)^n\gamma_ne^{x^2/2} \frac{d^n}{dx^n}e^{-x^2},
$$
where $\gamma_n=\pi^{-1/4}2^{-n/2}(n!)^{-1/2}$, be Hermite function.
Consider
$$
...
2
votes
0answers
50 views
Choosing boundary conditions for $(\frac{-d^2}{dx^2})^m$ on $H^m((0,1))$?
Consider the differential operator $D:$
$$
Du:=\frac{-d^2}{dx^2}u
$$
on the function space
$$
C=\{u\in C^2([0,1]):u(0)=u(1)=0\}.
$$
It's not hard to find the eigenvalues and ...
2
votes
0answers
133 views
Why is del operator coordinate free?
In solving Laplace equation $\Delta u=0$, every textbook will tell you to transform $\Delta=\partial_{xx}+\partial_{yy}$ into polar coordinate form $\Delta_p$(What it looks like doesn't matter here). ...
2
votes
0answers
453 views
Solving geodesic problems with Euler-Lagrange equation
This is the question:
Problem B.1 Two cities - Tel-Aviv, Israel and SanDiego, CA - have the same latitude 32
◦ N, but, different
longitudes: Tel-Aviv is 34
◦
E and San-Diego is 117
◦ W.
What is the ...
2
votes
0answers
119 views
PDEs with non-local terms
Not sure if I've used the correct terminology here (`non-local'). I think the lack of knowing the correct terminology is why I haven't been able to find any information about my query thus far.
I'm ...
2
votes
0answers
96 views
How to compute the monodromy group for a given differential equation.
Given a differential equation, say
$\frac{d^2f}{dz^2}+\frac{f}{z-5}=0$. We know that this equation has two linearly independent solutions $f_1(z), f_2(z)$. By analytic continuation, the solution ...
2
votes
0answers
388 views
Explain Triangle perimeter in polar coordinates
The question is to give a formula in $x$ and $y$ that gives all three sides of an equilateral triangle. The formula should not be true for points that are not part of the perimeter of the triangle. ...
2
votes
0answers
41 views
Showing uniqueness of solution to IVP under certain conditions
Consider the IVP $\mathbf{\dot x} = f(\mathbf{x},t)$ where $\mathbf{x}(0) = 0$, $f$ is continuous in some neighborhood of $(x,t) = (0,0)$ and $|f(x,t)-f(y,t)| \leq \frac{|x-y|}{t^\alpha}$. I would ...
2
votes
0answers
224 views
Lebesgue Line Integrals - Parametric Change of Variables
Consider the following Lebesgue integral in $\mathbb{R}^n$
$$ \int_C f(x) dx $$
Where $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is measurable and $C$ is a measurable subset of $\mathbb{R}^n$ that ...
2
votes
0answers
250 views
Conditions for bounded/periodical solutions of second order non-homogeneous ODE
Find all the possible values of $a$ and $b$, so that the equation:
$$
\ddot{x} + a\dot{x} + bx = \sin t
$$
Has only bounded solutions on $\mathbb{R}$
Has only periodical solutions
In general, we ...
2
votes
0answers
96 views
Positive rotational symmetric solution for p-Laplacian
I have the the following problem and I just can't get my head around how to solve it.
Be $1<p<n$ and $q=\frac{np}{n-p}$, $u\in\mathcal{C}_{n,p}=\{f\in W^{1,p}_{loc}: ...
2
votes
0answers
185 views
Grand Prix Race
Driver A has boon leading archrival B for a while by a steady 3 miles. Only 2 miles from the finish, driver A ran out of gas and decelerated thereafter at ta rate proportional to the square of his ...
2
votes
0answers
129 views
Where did G. W. Hill develop his relative motion equations?
The equations for relative orbital motion are commonly known as "Hill's equations" (also Clohessy-Wiltshire equations), and the citation given to G. W. Hill's 1878 "Researches in Lunar Theory" in the ...
2
votes
0answers
136 views
Implicit function theorem and consistency of a semi-explicit DAE
This may be a trivial question, but here goes:
Suppose a semi-explicit differential-algebraic equation (DAE) system is defined as follows:
$$
\begin{align}
&\dot x = f(x,z,\theta),\qquad x(0) = ...
2
votes
0answers
79 views
One question on 1st-order PDE
Given a smooth vector field $\mathbf{b}$ on $\mathbb{R}^n$, let
$\mathbf{x}(s)=\mathbf{x}(s,x,t)$ solve the ODE
$$\dot{\mathbf{x}}=\mathbf{b}(\mathbf{x}) (s\in\mathbb{R}), x(t)=x.$$
(a) ...
2
votes
0answers
233 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 ...
2
votes
0answers
579 views
Is there a strategy for solving a non-autonomous differential equation?
I'm curious about techniques for solving a nonautonomous* system in the case of a non-linear differential equation.
There's a simple example in my textbook (Hirsch, Smale, Devaney) where we obtain ...
2
votes
0answers
233 views
Which branch of mathematics is this and what are the introductory references?
I am self-studying a physics textbook on waves. While discussing solutions to linear homogeneous ODEs, the author talked about the exponential as "irreducible" solutions and on a footnote, said that ...
2
votes
0answers
69 views
Asymptotic stability of semi-trivial solution and existence of a nontrivial solution
Thank's amWhy!
I pray to some kind soul to help me on the theory of bifurcation: In the article of Tao Peng titled: "Bifurcation Behavior of a Cohen-Grossberg Neural Network of two Neurons with ...
2
votes
0answers
77 views
methods of solving differential/functional iteration equations
Let $f^{[n]}(x)$ be the $n$-th functional iterate of $f(x)$, so that $f^{[1]}(x)=f(x)$ and $f^{[n+1]}(x)=f(f^{[n]}(x)$. And let $f^{(n)}(x) = \frac{d^{n}}{dx^{n}} \left(f(x)\right)$
Has there been ...
2
votes
0answers
57 views
An estimate for the left point in a BVP
Let $\alpha \geq 1$.
Suppose that for each $c\geq c_0>0$ there exists a point $\xi (c) \in ]0,1[$ s.t. the BVP:
$$\begin{cases} [x^\alpha u^\prime (x)]^\prime +c\ u(x)=0 &\text{, in } ...
2
votes
0answers
385 views
Finding the modified Green function for the Helmholtz equation
I've been wrestling with this question for quite some time now, and the result was like 20 leaves of paper packed with scribbling...anyway, here's the question:
I need to find the solution to the ...
1
vote
0answers
23 views
What is the difference between an implicit ordinary differential equation and a differential algebraic equation?
I'm rather confused on this particular point. What is the difference between an implicit ordinary differential equation of the form:
x' = f(x',x,t);
and a ...
1
vote
0answers
17 views
Lipschitz constant for $F(t,y,z)=(z,f(y,z)\sin t)$
Let $f\in C^1(\mathbb R^2,\mathbb R)$.
Prove that all solutions for $x''=f(x,x')\sin t$ such that $x(0)=x(2\pi)$ and $x'(0)=x'(2\pi)$ have period $2\pi$.
I'm in the process of solving the above ...
1
vote
0answers
41 views
optimal control -Taylor expansion - PDE problem
I am trying to follow perturbation analysis in this paper (Optimal control of fluid limits of queuing networks and stochasticity corrections) and I am stuck at one point.
For the given control ...
1
vote
0answers
18 views
How to solve two-level Schrödinger equation using Floquet theorem?
Consider a sinusoidal driving two-level system:
$$
i \left(
\begin{array}{c}
\dot C_1(t) \\
\dot C_2(t) \\
\end{array}
\right)=\left(
\begin{array}{cc}
-\frac{\omega _0}{2} & ...
1
vote
0answers
21 views
Derivative methods for artifical neural networks with single hidden layer
I am trying to optimize the output of a given neural network with a single hidden layer. To accomplish this, I intend to find solve for all combinations of inputs where the derivative of the neural ...
1
vote
0answers
27 views
How do you set up a system of ODE's for this problem?
The problem is as follows:
Black and White balls are being created inside an arbitrary volume at rates of $Q_{B}$ and $Q_{W}$. They also disappear from the volume at rates $\lambda_{B}$ and ...
1
vote
0answers
29 views
Schroedinger equation in cylindrical coordinates
How can one numerically solve the nonlinear stationary Schroedinger equation in cylindrical coordinates?
1
vote
0answers
51 views
When is it justified to approximate a difference equation with its corresponding differential equation?
Consider the difference equation $f_{x+1}-f_x=a(f_x)$ and the differential equation $g'_x=a(g_x)$. When and Why is it justified to say "$f_x - g_x = o(1) $ hence we can solve the difference equation ...
1
vote
0answers
115 views
“Two-speed” linear integro-differential equation
Working on a problem of many-electron dynamics in quantum dots I have arrived to an a following integro-differential equation:
$$\frac{\partial}{\partial t} F(x,t)= - i (x+ v_1 t) F(x,t)-\alpha^2 ...
1
vote
0answers
26 views
Invariant relation in ODE
It is well known that if function $g(x)$ is an invariant relation under ODE $\dot x = f(x)$ then $\frac{\displaystyle d}{\displaystyle dt}g = \lambda g$.
More precisely. Let ...
1
vote
0answers
30 views
A Nonzero Alternating Bilinear Form on the Space $P_1(F)$ Over $F$
Can anybody think of an example of a nonzero alternating bilinear form on the space $P_1(F)$ over $F$.
$F$ is a general field like $\mathbb{R}$ or $\mathbb{C}$.
$P_1(F)$ is the set of all ...
1
vote
0answers
48 views
Approximating the modified Bessel’s function with a sum of exponentials
I am looking for an approximation for modified Bessel’s function $I_\alpha(f(t))$ (specially $I_0(f(t))$ or at least $I_0(t)$) with a sum of exponential functions. I mean I want to approximate the ...
1
vote
0answers
41 views
Unusual jump condition for Green function
This question is related to a previous question I posted a while ago.
Imagine that I'm computing the Green function of a linear operator $L$, such that:
$$LG(x,s)=\delta(x-s).~~~~~~~~~~~(1)$$
Now, ...
1
vote
0answers
21 views
Green's function, stuck on this particular equation
How does one find the Green's function for a differential equation in two variables which looks like,
$\frac{\partial}{\partial t} P -\omega P = j$
where $\omega$ is a $2\times 2 $matrix with ...
1
vote
0answers
36 views
How do I solve an optimal control problem when the state and control are multiplied?
Suppose I have the following objective function
$$
R = \sum_t^T x_tu_t + ku_t^2
$$
subject to
$$
\Delta x_t = m u_t + n x_t
$$
where $x_t$ is the state and $u_t$ is the control. $x_0$ is known.
How ...
1
vote
0answers
14 views
How to find variance of a CIR process
CIR process is defined as follows:
http://en.wikipedia.org/wiki/CIR_process
I get an SDE form for d_Vt/dt, but can't proceed further.
1
vote
0answers
38 views
Pull Back (change of variables)
Let be $h:\mathbb{R^2}\rightarrow\mathbb{R^2}$ a change of variables (diffeomorphism). Let be $X$ a vector fields in $\mathbb{R^2}$ and $f:\mathbb{R^2}\rightarrow\mathbb{R}$ a continuous application. ...
1
vote
0answers
31 views
First eigenvalue of the given linear operator
I have the following question:
Let us denote $H_2^N: = \{u\in (H^2(0,1))^2: u'(0) = u'(1) = 0\}$.
Let an operator $L:H_2^N \to (L^2(0,1))^2$ be given by
$Lu = -Du'' + Cu$, where $D$ is a positive ...
1
vote
0answers
20 views
zero stability of odes
Can anyone help with the following problem?
Find the range of α for which the following method is zero stable.
$$y'=f(x,y)\\
f(a)=y_0 \\
y_{n+1}-(1+α) y_n + α y_{n-1}=\frac{1}{2}h \left[(3-α) ...
1
vote
0answers
24 views
Solve I.V.P for differential using quadratic form
Solve the i.v.p for $y''+4y'+5y=0, y(\frac{\pi}{2})=1/2, y'(\frac{\pi}{2})=-2$
I solved using the quadratic form. and I got $\lambda = \frac{(-4 \pm 2i)}{2}$, which for $\lambda 1,2= 2+2i$.
And then ...
