Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

learn more… | top users | synonyms (1)

1
vote
0answers
35 views

Trace of a tensor in a differential equation

If $Z$ is a rank-2 tensor, does the following differential equation mean anything to anyone: $\nabla^2Z+\frac{1}{c^2}\frac{\partial^2}{\partial t^2}tr(Z)=0$ The presence of this trace really blurs ...
1
vote
0answers
33 views

Change of variables FPE

Given the partial differential equation: $$\partial_tP(z,t)=-\partial_z[(-z^2+A)P(z,t)]+D\partial_{zz}P(z,t)$$ where $A$ and $D$ are constant parameters. how to remove $z^2$ term by substitution?
1
vote
0answers
18 views

Sequence of periodic orbits in a bidimentional vector field

Let $X:\Delta\longrightarrow \mathbb{R}^2$ a vector field and $\gamma\subset\Delta$ a periodic orbit of $X$ with periodicity $\lambda$. Let ${\gamma_n}$ a sequence of closed orbits of $X$ with ...
1
vote
0answers
29 views

help with a differential equation

I have a differential equation of the form $$f(y)\dfrac{\mathrm{d}y}{\mathrm{d}z} + g(y)=h(z)+K$$ with $K$ constant. I need to know, what kind of equation is it? and how I can solve it? can be ...
1
vote
0answers
76 views

Lipschitz dependence on ODE parameters in discrete setting

Consider the ordinary differential equation $$ x'(t) = e^{-x(t)} - p$$ with (time-independent) non-negative parameter $p$. Its solution after time $t$ is given by $$ \Phi^t_px_0 = \log\left( e^{x_0 - ...
1
vote
0answers
35 views

Solutions for ODE with periodic B.C.

I have the following ODE $$-u(x)''=f(x) \qquad u:[0,1)$$ It is smooth and 1-periodic. Assume that I have a solution u(x). How do I prove that: $$u(x)= u(x)+c \quad c \in \mathbb{R} $$ is also a ...
1
vote
0answers
70 views

$o(|X(t+\Delta t)-X(t)|) $ is $o(\Delta t)$?

X is differentiable. if $X'(t) \neq 0 $ , it's easy to show. $ \lim_{\Delta t \to 0} \frac{o(|X(t+\Delta t)-X(t)|)}{|X(t+\Delta t)-X(t)|} \frac{|X(t+\Delta t)-X(t)|}{\Delta t} = 0 \cdot |X'(t)|$ ...
1
vote
0answers
74 views

Lyapunov's stability

I'm working on the following differential equations: $$\frac{dN}{dt} = wN(1-\frac{N}{p})$$ $$\frac{dK}{dt} = sK - gKN$$ where $w, p, s, g$ are real numbers $\ge 0$ and $N, K$ are always positive (or ...
1
vote
0answers
300 views

Hamiltonian and Gradient System Phase Planes and Level Curves

I am given a function $H(x,y)=ysin(x)$ which is Hamiltonian. I want to sketh the phase portrait and level curves for the Hamiltonian system as well as the corresponding gradient system. Then the ...
1
vote
0answers
200 views

How to integrate functions involving absolute value?

$\displaystyle b_n = \frac{2}{10}\int\limits_{0}^{10} 4 - 0.8 \vert x - 5 \vert \bigg(\frac{\sin(nx\pi)}{10}\bigg) \, dx$ This is in regards to my previous question, which I have deleted due to my ...
1
vote
0answers
253 views

Second Series solution y(2) for Frobenius Method

I am currently solving the Frobenius Method for the question $xy'' +y = 0$ given the ICs $y(0) = 0, y'(0)=1$ I have done some work into solving that the first series solution for $y_1 = ...
1
vote
0answers
34 views

Solution of equation with special function using maths

I have the following equation with β∈[0,1] and δ∈[0,1] are 2 parameters and $\mu$, $\sigma$ are the mean and standard deviation of a random variable. Is it possible to use software to get the explicit ...
1
vote
0answers
66 views

Eigenfunction Expansion for Simple Nonhomogenous PDE

How do I go about finding an eigenfunction expansion for the following equation: $$ u'' = f(x)$$ where: $$ u'(0) = \alpha \quad u'(1) = \beta$$ What about the case when $f(x) = C$ a constant? Edit: ...
1
vote
0answers
44 views

Checking if a function is in the Schwartz space of rapidly decreasing functions.

Is there any neat bi-implication other than the definition that I can use to check this? This question was motivated by a question that asked if $ f(x) = e^{-|x|^3}$ was in S. It isn't infinitely ...
1
vote
0answers
65 views

Neumann boundary condition for smooth function defined on the interior

Let $\Omega\subset\mathbb{R}^n$ be open and let $f\in C^\infty(\Omega)$ be a smooth function. What examples can one come up with that distinguish the 3 criteria below? 1: f satisfies the Neumann ...
1
vote
0answers
185 views

Plotting trajectories of a 2x2 nonlinear autonomous system

As shown in Boyce and Deprima (10th Ed p. 515), one way to model the trajectories of the solution to the system of differential equations as defined below is in the case where we can find ...
1
vote
0answers
76 views

Smoothing operator on manifolds

I am reading John Roe "Elliptic operators, topology and asymptotic methods". On page 79 there is the definition 5.20 of smoothing operators. The definition is the following: "A bounded operator on ...
1
vote
0answers
65 views

Difference Equations and displacement operator

For a Prep exam Exercise from the book: Numerical analysis of scientific computing. Section 1.3-3 Let $p$ be a polynomial of degree $m$, with $p(0) \neq 0$. If a sequence $x$ contains $m$ ...
1
vote
0answers
81 views

Modified Green's function

The question is find the modified Green's functions for $$u''+u=f(x)$$ with boundary conditions $$u(0)=0, u(\pi)=0$$ I know I should modified the Green's function as ...
1
vote
0answers
71 views

A singular boundary value problem

Is there any numerical approach to solve a BVPs for ODEs of the form: $y'=\frac{ky^2-y^{3/2}-y}{\beta t}$ with initial point $(0,y0)$? I know a problem of the form $y' = \frac{S}{t}y+f(t,y)$ with ...
1
vote
0answers
44 views

How do I determine whether the image of a function lies in an algebraic curve?

On p. 2 of the book Differential Equations and Dynamical Systems by Lawrence Perko we define the function $$x(t) = \left(c_1 e^{-t}, c_2 e^{2t}\right)$$ It is then stated that the curve defined by ...
1
vote
0answers
77 views

Solution of differential equation with polynomial coefficients

I have come across a rather general form of differential equation and I was wondering if anyone knew what they would be called or where I can find some literature on them. Imagine we have some ...
1
vote
0answers
77 views

Solution of linear PDE

I want to solve linear equations, which can be found in the article $\rho_t+\dfrac12[\rho(U-v\cos\theta)]_x=0,$ ...
1
vote
0answers
146 views

application of rectification theorem on first integrals

Be $X:\Delta\longrightarrow \mathbb{R}$ a vector field of class $C^{1}$ where $\Delta\subseteq \mathbb{R}^n$ is open. We say that $f:\Delta\longrightarrow\mathbb{R}$ is a first integral of $X$ in ...
1
vote
0answers
51 views

Probability of winding number in 2D Brownian motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...
1
vote
0answers
200 views

How do you numerically solve a multivariable ODE system with different time steps per state variable?

If you have a large multivariable ODE system, and certain processes occur at a much smaller time scale, how can you implement a solver that uses smaller time steps for state variables involved in fast ...
1
vote
0answers
40 views

How can I prove this theorem about differential inclusions?

Consider the following differential equations with initial conditions at time $t_0$ specified: $\dot{x}_1 = f_1(x_1,t) ; \,\,\,x_1: [t_0,T]\to\mathbb{R}^n, ...
1
vote
0answers
107 views

What is the solution of the ODE for the quadratically damped harmonically driven oscillator, e.g. the linearized pendulum in air.

There are many sources for the driven linear damped and even many for the quadratically and constant damped regimes of free (not driven) ODE solutions, but more useful is the quadratically damped and ...
1
vote
0answers
65 views

Solve using Riccati equation?

In this problem, they've asked to solve it using the solution $y=-\frac{1}{3x}$ but the problem is that the solution just makes it more complicated. The equation which needs to be solved is, ...
1
vote
0answers
97 views

How to show that the first derivative is bounded in a function

\begin{equation} y=\arccos\left( -\frac{1}{2\left(Dr^{\dfrac {|\sin(2x+\theta)|}{M\sin x\sqrt{A+2B\cos(2x+\theta)}}}+1\right)} \right) \nonumber \end{equation} How to show in above function the ...
1
vote
0answers
94 views

1D PDE decoupling

I need to solve the 1D nonlinear poisson equation and I thought of trying the fixed point decoupling technique. The equation is this: $\frac{\partial^2 \phi(y)}{\partial ...
1
vote
0answers
70 views

Particle in a Polya Vector field

For a given analytic function $H$ from $\mathbb{C}$ to $\mathbb{C}$, we define the Polya Vector Field to be $\bar{H}$. This then corresponds to a irrotational, conservative vector field on ...
1
vote
0answers
316 views

Show that Bellman-Gronwall's inequality

I'm trying to prove the theorem general of the Bellman-Gronwall's inequality: Assume that $u(t)$ be real valued non - negative continuous function, and such that $$u(t)\le u(\tau ...
1
vote
0answers
145 views

Find the smallest interval you can be sure contains $g(1.8)$.

Let $g$ be a function such that $g(2)=1$, $g′(2)=2$ and $|g′′(x)|<0.4+(x−2)^2$ for all $x>0$. Find the smallest interval you can be sure contains $g(1.8)$. Your answer should be the endpoints ...
1
vote
0answers
49 views

solvability of specific differential equations

I've come across a type of differential equation and I'm wondering if it can be solved, or if there are special conditions on the equation in order that it is solvable. The equation can be written ...
1
vote
0answers
121 views

Is there a generalization of the ODE Comparison Theorem to n dimensional systems such as this one?

Is the following theorem true? If so, under what conditions? If not, why not? For any finite set of points $S$, let $conv(S)$ denote the convex hull of $S$. Let $f:\mathbb{R}^{n+1} \to \mathbb{R}^n ...
1
vote
0answers
30 views

Getting Eigenvalues Into a Differential Operator?

Following Butkov, a second order ode $$A(x)y'' + B(x)y' + C(x)y = D(x)$$ can always be brought into Sturm-Liouville form $$\tfrac{d}{dx}[p(x)y'] - s(x)y = f(x)$$ after multiplying across by ...
1
vote
0answers
113 views

How to find the Frobenius series solution of $\cos x~y''+xy'+by=0$?

How to solve the equation in series form $$ \cos{x}~y''+xy'+by=0 $$ where b is a real constant? Here is what I tried: $$ \cos{x}=\sum_{m=0}\frac{(-1)^mx^{2m}}{(2m)!} $$ $$ y=\sum_{n=0}a_nx^{n+s} ...
1
vote
0answers
66 views

Solving systems of differential algebraic equations: Is it legitimate to hold some variables constant?

I have a system of linear differential and algebraic equations, along with some non-linear equations. That is, I have a system of equations that can be written in the form $\mathbf{A}y'(t) + ...
1
vote
0answers
239 views

L-stability and Stiff decay

In my Numerical Methods for PDEs textbook by Ari Uscher L-stability and stiff decay are introduced by considering a generalized test equation: $y' = \lambda (y - g(t)), 0 < t < b$ where g(t) ...
1
vote
0answers
103 views

Generic invertibility of a Gramian-like matrix

Motivation: Consider the cascade arising from connecting the system $\dot{x}_1 = A_1 x_1 + B_1 u_1$ with the system $\dot{x}_2 = A_2 x_2$, $y_2 = C_2 x_2$ according to $u_1 = y_2$, namely: $\dot{x} = ...
1
vote
0answers
59 views

A method called “incorrect method”

Good night. Is there a method called "incorrect method" to calculate second order differential equations? If so, please, is there a web page about it, as I have to investigate this method? Thank ...
1
vote
0answers
45 views

Existence results for this ODE? (periodic)

Are there any existence/uniqueness results for solutions to the ODE $$y'(t) = f(y(t),t)$$ $$y(0) = y(T)$$ on the time interval $[0,T]$ where $f$ is Caretheodory and $T$-periodic in $t$. I am looking ...
1
vote
0answers
100 views

A system of nonlinear partial differential equations

Here are non-linear partial differential equations, where $f$ and $g$ are functions of $x,t$ : $g^2 (\partial_{x}f)(\partial_{t}f) - (\partial_{x}g) (\partial_{t}g) = 0, \quad ...
1
vote
0answers
108 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) ...
1
vote
0answers
81 views

one dimensional differential equation coordinate transformation

I need to show that there exist a coordinate transformation $x\longmapsto y=x+\delta x^2$, with $\delta$ constant, such that the following ODE $$\dfrac{\text{d}x}{\text{d}t}=\sum_{i=1}^{n}{a_ix^i} ...
1
vote
0answers
61 views

A problem in differential equation

I am trying to solve following problem. If you have an idea or a solution, I would really appreciate it. Let $a : \mathbf{R} \rightarrow \mathbf{R}$ be a $T$- periodic continuous function and $x(·)$ ...
1
vote
0answers
92 views

Reflection Principle

$U^+ \colon= \left\{x\in \mathbb{R}^n\mid |x| < 1, x_n>0\right\}$ is an open half-ball. Assume $u \in C^2 (\overline{U^+}$) is harmonic in $U^+$ with $u=0$ on $\partial U^+ \cap \{x_n=0\}$. ...
1
vote
0answers
187 views

Non-linear ODE for which backward Euler becomes unstable?

One way to solve initial value problems of the type $\dot{x} = f(x), \; x(0) = 0$ numerically is to use the backwards Euler method $x_{n+1} = x_{n} + \Delta t f(x_{n+1}), \; n = 1,\ldots \\ x_{1} = ...
1
vote
0answers
1k views

Differential Equation Birth-Death Rate Model

I'm working on the following problem: Birth and death rates of animal populations typically are not constant; instead, they vary periodically with the passage of seasons. Find P(t) if the population ...