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

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142 views

finite difference equations

i havent had a response to this question in a while, could someone please help me. Im struggling to understand the concepts of forward/backward/central differences on finite difference equations. i ...
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0answers
129 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 ...
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1answer
102 views

What is the time-integral of motion for first order differential equations?

For a second order differential equation (many physical systems) in one variable, I know "procedures" to compute the energy. Given $$q''(t)=f(q(t),q'(t)),\ \ q(0)=q_0,\ \ q'(0)=v_0,$$ if we're lucky ...
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1answer
182 views

Solution of a differential matrix equation

Given a differential matrix equation, ie $X'=A(z)X+B(z)$ where both $A$ and $B$ are matrix of size $n\times n$ with coefficients that are holomorfic functions in a convex open set $\Omega$ and ...
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2answers
744 views

Matrix Differential Equation with a Skew-Symmetric Matrix

From a bank of masters exams: Say the position of a particle moving in $\mathbb{R}^n$ is given by a smooth vector-valued function $\vec{x}(t)$. Suppose that $\vec{x}(t)$ satisfies a ...
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3answers
351 views

Express differential equations as system of first order equations

Express the differential equation $$y'''-6y''-y'+6y=0$$ as a system of first order equations i.e. a matrix equation of the form $$A(\vec x)'=0$$ where $$\vec x\text{ is the vector }\left[ ...
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1answer
653 views

Isolated Versus Non-Isolated Fixed Point, 2D Dynamics

I am trying to understand the classification of fixed points in a dynamical systems context (fixed points of a system of two linear differential equations are places where both $x_1' = x_2' = 0$). ...
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1answer
294 views

Solving ODE using frobenius method. 3 coefficients

I'm trying to learn frobenius method by solving some problems (ODEs). For example: $$xy''+(2x+1)y'+(x+1)y=0$$ Let $y=\sum\limits_{n=0}^\infty a_nx^{n+r}$. Then, I took derivatives and put into the ...
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1answer
96 views

Differential equations basic problem

I know this is a basic Physics problems but somehow I can't solve it. We have the differential equation: $2x''x^2 - 4 x^2x' - 2 x^3 = 0$ We have to conclude that the system: $x' = y $ $y' = 2y + ...
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1answer
43 views

Existence of Phase Flow

Hi there I'm wondering if anyone can clear up my confusion: What is the proof of the local existence of a phase flow for a differentiable vector field?
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1answer
125 views

How to make a unit step function?

I am trying to make a unit step function. I have this function (the equation of an ellipse, not centered at the origin): $$ f(x,y) = \frac{(x-X_c)^2}{a^2}+\frac{(y-Y_c)^2}{b^2} $$ What I would ...
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1answer
128 views

minimization problem on differential equations - optimal control

I am trying to minimize an time-integral of a linear function with respect to differential equations. The problem is formally defined as follows: Given $\lambda< \mu_1, \mu_2$ fixed ...
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1answer
109 views

continuity and differentiability and L'Hopital's Rule [closed]

Let $$f_n(x) = \begin{cases} 0 & x < -\tfrac{1}{n} \\ \tfrac{n}{2} & -\tfrac{1}{n} \leq x \leq \tfrac{1}{n} \\ 0 & x>\tfrac{1}{n} \\ \end{cases},$$ $n=1,2,3,\ldots$. Let $g(x)$ be a ...
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0answers
35 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 ...
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1answer
28 views

System of Differential Equations Question Assistance

The following question has just left me confused with no real decent avenue of attack so any assistance on this would be appreciated. For the system of equations $t {\frac{d \vec x}{dt}} = A\vec x $ ...
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1answer
293 views

Are these ODEs equivalent?

I have the following set of ordinary differential equations: \begin{equation} \left\{ \begin{array}{l} \dot{a} = f_1(a, b, c, d) \\ \dot{b} = f_2(a, b, c, d) \\ \dot{c} = f_1(c, d, a, b) \\ \dot{d} = ...
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0answers
50 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 ...
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0answers
106 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 ...
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1answer
1k views

Rewrite matrix equation for Euler method and Improved Euler method

Consider a system of the form: (1) $x' = Ax + g$ For appropriate matrices $x'$, $A$, $x$, and $g$. If we let $y_n$ be the approximation to the solution of (1) at time step $t_n$, what matrix ...
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0answers
130 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, ...
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1answer
2k views

Euler's method for second order differential equation

Not really homework but sample exam. The question is to use Euler's Method to approximate Y: $Y''(t) = Y'(t) - 2Y(t)$ $Y'(0) = Y(0) = 1$ with $t_0 = 0$ and $h=0.2$ So what I did: First ...
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2answers
789 views

stability and asymptotic stability: unstable but asymptotically convergent solution of nonlinear system

Consider nonlinear systems of the form $X(t)'=F(X(t))$, where $F$ is smooth (assume $C^\infty$). Is it possible to construct such a system (preferably planar system) so that $X_0$ is an unstable ...
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1answer
119 views

Simple Diffy-Q problem

So as a fun project, I'm trying to work my way through Kreyzig's "Advanced Engineering Mathematics". But I've gotten to a really simple problem: $$xy' = 2y$$ where I know the solution is $x^2$ but ...
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1answer
23 views

If $\lim_{t\to\infty}\gamma(t)=p$, then $p$ is a singularity of $\gamma$.

I'm trying to solve this question: Let $X$ be a vectorial field of class $C^1$ in an open set $\Delta\subset \mathbb R^n$. Prove if $\gamma(t)$ is a trajectory of $X$ defined in a maximal ...
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1answer
118 views

Bifurcation value and description

Find the bifurcation of $a$ and describe the bifurcation that take place at each value $\displaystyle dy/dt=e^{-y^2}+a$ I let $\displaystyle e^{-y^2}+a=0$ then solve for y. I got $y^2=-\ln(a)$ What ...
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1answer
82 views

Better than Runge-Kutta-Fehlberg 4(5) at high order?

I wonder what are currently the best numerical solvers of ODE for high-accuracy computations. I need an efficient and accurate method to solve ODE that are not pathological (all is smooth) using ...
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2answers
64 views

Does this ODE have an exact or well-established approximate analytical solution?

The equation looks like this: $$\frac{\mathrm{d}y}{\mathrm{d}t} = A + B\sin\omega t - C y^n,$$ where $A$, $B$, $C$ are positive constants, and $n\ge1$ is an integer. Actually I am mainly concerned ...
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1answer
40 views

How can I prove that the extremes of the interval of the solutions of this differential equation are equilibrium points?

I'm trying to proof if $x:I\to \mathbb R$ a maximal regular solution of $x'=f(x)$, such that the image $x(I)\subset \mathbb R$ is bounded and $f:\mathbb R\to \mathbb R$ is $C^1$, then the extremes $a$ ...
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1answer
27 views

How do we know which component belongs to which part in a separable differential equation

Take for instance dP/dt = kP We get after separating: dP/P = kdt, but why shouldn't it be dP/kP = dt instead, mathematically it doesn't make sense to say that k must belong absolutely to the right ...
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1answer
41 views

terminology for the derivative of a trajectory in phase space

Suppose we have 2nd order ODE. For example: $x'' + x = 0.$ We can view this as a first order ODE in two dimensions: $x' = v$ $v' = -x$ What is the vector $(x', v') = (v, -x)$ called?
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1answer
111 views

Differential equation model and graph question

Consider the model $dS/dt=f(S)= kS(1-S/N)((S/M)-1)$ Where S is the population of the squirrel, k and M are constant parameter, N is the carrying capacity. Know that the more people move in, the more ...
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1answer
38 views

can we divide by any term when we have an differential homogeneous equation?

I am asking because i think we divided by x here for whatever reason since the other side is equal to 0 and it wont affect the equation in any meaningful way. I got x/2 = y + c instead The thing I ...
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1answer
74 views

Generalisations of the Gronwall's lemma

Suppose we have the following differential inequality $F''(w)\le \frac{p-1}{p}\frac{(F'(w))^2}{F(w)}$ on $w\in(w_0,w_0+\varepsilon)$, $w_0>0$, $\varepsilon>0$, $p>1$. In addition, ...
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1answer
144 views

Combining differential equations

Can anyone see how to combine the following 3 equations $$\dot r^2-\dot\theta^2=-\theta^2$$ $$\theta\ddot \theta-2\dot \theta^2=2(\dot r^2-\dot \theta^2)$$ $$\dot r=a \theta^2$$ to get ...
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1answer
657 views

Possible ways to do stability analysis of non-linear, three-dimensional Differential Equations

For example Lorenz system, $$ \frac{d}{dt}\begin{pmatrix} x\\ y\\ z \end{pmatrix}=\begin{pmatrix} -\sigma & \sigma & 0\\ \rho & -1 & -x\\ y & 0 & -\beta ...
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1answer
41 views

Scale free property of Pareto distribution

I am trying to show that the Pareto distribution is scale free, defined as: p(bx) = g(b)p(x) I get to this stage: ...
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1answer
55 views

Can't do this ODE question

http://i.imgur.com/6NMcPdD.jpg I have an exam in 2 hours, this question I cannot get my head around. So far I managed to get the first order and second order differentials for $z(t)$ And found that ...
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2answers
319 views

How to verify the order of DOPRI Runge-Kutta method

I've written code in Fortran based on the RK8(7)-13 method by Dormand and Prince to solve the system $\mathbf{y}'=\mathbf{f}(t,\mathbf{y})$. The method is ...
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1answer
56 views

Differential linear equation first order?

I have the equation $(1+y^2)\,dx+(xy+1)\,dy=0$. It is a linear differential equation of the first order. Now the problem is that this doesnt have the regular form of this type of equations which is ...
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1answer
62 views

Explanation needed for problem related with differential equation $w''(x)-q^2 w(x)=0.$

I am stuck on the following problem: Suppose that $q \in \Bbb C.$ Consider the differential equation $w''(x)-q^2 w(x)=0.$ If every solution of this equation satisfies $$\sup_{T >0} ...
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1answer
66 views

Systems of Differential Equations and higher order Differential Equations.

I've seen how one can transform a higher order ordinary differential equation into a system of first-order differential equations, but I haven't been able to find the converse. Is it true that one can ...
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1answer
85 views

General solution and intervals (over which the general solution is defined)

So I did a homework and I got x^2/(x+1)e^x + C/(x+1)e^x for all R (except -1), but they say it's for -1 < x < oo+. I don't understand how that makes any sense. Is there something I am not ...
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2answers
40 views

Solve $y'+a(x)y=b(x)$ where $b(x)$ is not continuous

Find all the solutionsof the equation: $$y'+ay=b(x),\ 0<x<\infty,\ $$where $a$ is a constant and $b(x)=1$ for $0\le x\le \alpha$, and $b(x)=0$ for $x\gt \alpha$ and $\alpha$ here is a ...
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0answers
74 views

How to solve a system of differential equations

When I solved one problem, I faced with the need to solve the following system of differential equations: 1) $ \ddot{x}(t)-a(t)x(t)-b(t)y(t)-c(t)=0 $ 2) $ \ddot{y}(t)-d(t)y(t)-b(t)x(t)-e(t)=0 $ ...
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1answer
993 views

bifurcation value

I tried to understand how to locate the bifurcation value for the one-parameter family. From my understanding the bifurcation value is the maximum or minimum point of a parabola, so I set the ...
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1answer
60 views

Examples of systems conforming the Lorentz Attractor

Might sound like a trivial question but would you please show me some examples of real systems conforming the Lorentz Attractor? It can be any kind of system, just a little list. It can be a system ...
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0answers
108 views

A photon in expanding Universe (a snail on a tree)

I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...
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2answers
57 views

Bernoulli differential equation help?

We have the equation $$3xy' -2y=\frac{x^3}{y^2}$$ It is a type of Bernouli differential equation. So, since B. diff equation type is $$y'+P(x)y=Q(x)y^n$$ I modify it a little to: $$y'- \frac{2y}{3x} ...
2
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1answer
141 views

Canonical Form and DE

If we have the differential equation $x'' = x - \cos(x')$, then In part a) Compute the corresponding non-linear 2D system and and its (unique) equilibrium part b) compute the linearized system at ...
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1answer
131 views

How can I prove that the solutions of this differential equation is monotone?

I'm trying to proof if $x:I\to \mathbb R$ a maximal regular solution of $x'=f(x)$, such that the image $x(I)\subset \mathbb R$ is bounded and $f:\mathbb R\to \mathbb R$ is $C^1$, then $x$ is strictly ...