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

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Existence Theorem for Geodesics

The text I am reading now defined geodesics to be those curves that satisfy the following differential equation: $\ddot{\gamma}^k(t)+\dot{\gamma}^i(t)\dot{\gamma}^j(t)\Gamma^k_{ij}(\gamma(t)) = 0$ ...
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147 views

The simplest delay differential equation

I am trying to understand a bit about solutions of delay differential equations, so I tried analyzing one of the most simple ones: $$u'(t)=-\beta u(t-1), \text{and for } t\in [-1,0), u(t)=\phi(t), ...
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58 views

Second order equations on manifolds

In my notes, the lecturer considers a smooth vector field $v: TM\to T(TM)$, with $M$ a smooth manifold. Let's write $$v(u,e)=((u,e), (a(u,e),b(u,e)).$$ It is said that $v$ is a second order equation ...
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Behaviour of $r'=r-r^3 , \theta'=(\sin\theta)^2+a$

What are the local and global behavior of solutions of $r'=r-r^3$ $\theta'=(\sin\theta)^2+a$ at the bifurcation value $a=-1$?
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275 views

Integrating angular velocity to obtain orientation

Suppose that $\gamma:[0,1]\to \operatorname{SO}(3)$ is a path in the space of orientation preserving rotations of $\mathbb R^3$. It is classical that we can find a corresponding $\omega:[0,1]\to ...
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Complex nonlinear differential equation

I have the following nonlinear differential equation: $$\ddot z(t)-\sin(z(t))=0$$ where $z(t)$ is a complex variable. The solution of the same equation with $z(t)$ real, is a function of Jacobi ...
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247 views

system of implicit nonlinear differential equations

Here I have a system of nonlinear differential equations: $ (M+2m)\ddot{x} + m(l_1 \ddot{\theta}_1\cos\theta_1 - l_1\dot{\theta}_1^2\sin\theta_1) + ...
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253 views

Chebyshev Diff EQ

Find a power series solution about $x_0=0$ for the Chebyshev differential equation $$(1-x^2)y''-xy'+n^2 y=0,$$ as a function of of the integer $n$. Show that the solutions form a terminating ...
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103 views

The linearization of a gradient vector field along a heteroclinic connection

A gradient vector field $X$ in $\mathbb{R}^n$ has two equilibria $x_1, x_2$. The vector field defines a cooperative dynamical system. The linearization about $x_1$ has one positive eigenvalues and ...
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86 views

What's this called? $\mathbb{C}[d/dx]$

The 'ring of differential operators wrt x' ? Thx.
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234 views

Harmonic oscillator with stochastic forcing

It's well known that the solution of the differential equation: $$\ddot x(t)+\omega^2x(t)=\sin(\psi t)$$ has the form: $$x(t)=C_1 \sin(\omega t)+C_2 \cos(\omega t)-\frac{\sin(\psi ...
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Is it possible to have Wronskian=0 with independent solutions to a linear differential equation?

In Wikipedia it says that if the Wronskian of two function is 0 everywhere it does not imply they are linearly dependent. However, in books treating differential equations it seems that, if the two ...
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133 views

Prove there are at least two periodic solutions

Could anyone comment on the following ODE problem? Thank you. Given a 2-d system in polar coordinates: $$\dot{r}=r+r^{5}-r^{3}(1+\sin^{2}\theta)$$ $$\dot{\theta}=1$$ Prove that there are at least ...
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349 views

Gompertz growth equation

:) Hi! I'm almost finished with a homework problem, but I cannot quite finish it. The problem is as follows: Given the Gompertz growth equation $$\frac{dN}{dt}=K(t)N(t),\ N(0)=N_0 \\ ...
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698 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 ...
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135 views

Satisfying a Differential Equation and complex Laguerre

I have the following problem Show that $$L_n(x)=\frac{e^x}{2 \pi i}\oint \frac{t^n e^{-t}}{(t-x)^{n+1}}dt$$ satisfies $$x\, L_n^{\prime\prime}+(1-x)L_n^\prime+n\, L_n=0$$ where the contour is ...
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334 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 ...
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90 views

Steady-state of `degenerate' delayed differential equation

Consider the simple delayed differential equation: $X'(t) = -a X(t) + a X(t - d)$ where $d$ and $a$ are positive constants. I'm interested in the possible steady-state (stationary) solutions of ...
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212 views

Approximating a system of differential equations as a Bézier curve

I am looking for a general transform to approximate the solution to an n-dimensional system of differential equations and initial conditions as a cubic or quadratic Bézier curve. Sorry if my ...
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95 views

Finding $\mathbf r(t)$ for the parameterized two-body equations of motion

I'm trying to understand the equations of two-body motion. Namely, given the position, velocity and mass of two orbiting bodies at time $t$, how can I explicitly find their position and velocity for ...
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302 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 ...
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Special forms of ODEs

In my previous question, @Gerben suggested that it is more likely that WA recognizes an ODE in"Sturm-Liouville" form. Is there a reason for this particular form being preferred to the usual ...
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126 views

Inequality of ODE solutions

Says I have two (scalar) ODE: $u' = f(u,t)$ and $v' = g(v,t)$ where Both $f$ and $g$ are piecewise-continuous and locally Lipschitz, for existence & uniqueness of solutions $u(t)$ and ...
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Differential equation with some constraints

I'd like $\alpha,\beta,\gamma$ as functions of $t$, satisfying the following conditions: $$ \begin{align} \alpha+\beta+\gamma & = 0 \\ \sin^2\alpha + \sin^2\beta + \sin^2\gamma & = c^2 \\ ...
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235 views

Half-life versus relaxation time

Question: What is the exact relationship between half-life and relaxation time? I just wanted to nail down the difference/similarity between these two concepts. I did a web search, and even found a ...
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224 views

Series of nested double integrals

This is kind of a follow-up of my previous question. I'm investigating the following infinite series of nested two-dimensional integrals $$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 ...
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446 views

Differential Equations of Infinite Order

As a physicist I was playing with some QM problem and stumbled upon an ordinary differential equation of infinite order (coefficients are polynomials) that could be cast in the form: ...
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504 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 ...
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19 views

Is ODE theory useful for developing numerical solvers for ODEs?

I will be doing research in developing numerical solvers for ODEs. I was wondering if knowledge of ODE theory will be useful and if so in what ways. I am asking because, I am inclined to take a ...
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Adding a delta function to a differential equation

So say I have a differential equation of the form: $$ \left(\alpha \frac{d^2}{dx^2}+fx^2 \right)y(x)=\lambda y(x) $$ Whose solutions are known (a Gaussian multiplying a Hermite polynomial.) I am now ...
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second order differential equation with Green's function

I need to solve following differential equation \begin{eqnarray} y''(x) - k = \delta(x-x_0) \end{eqnarray} subject to conditions: \begin{eqnarray} y(x=-a) = 0 \\ y(x=b) = p \end{eqnarray} Is it ...
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using ode45 for descriptive forms

Using Matlab what would be the most efficient way to solve, $A_1x'(t) = A_2x(t)$, where both $A_1$ and $A_2$ are $n\times n$ matrices. Both are sparse matrices and hence I want to avoid inversion. ...
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Finding a solution basis of differential equation

Find a solution basis of $$y'=\left[ \begin{matrix}3&-4&-2\\2&-3&-2\\0&0&1\\ \end{matrix} \right]y \,\text{ and find the solution } \Phi \text{ with } \Phi(0) = (1,1,1).$$ I'm ...
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Quaternion conversion

We have a normalized orthogonal co-ordinate frame travelling through the curve as in figure 1 below, from one end to other. Let us call starting end as A and ending end as B. What we know is initial ...
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Special properties of bounded functions

I have a problem understanding the reasons as to why under some circumstances a term can be omitted due to it being a part of a bounded function, and I hoped to get some clarity to this here. There is ...
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16 views

Growth condition in differential equation and vanishing solution at boundaries

In a discussion on solving a partial differential equation I lately read: "Under a standard growth condition on the solution at infinity, the resulting PDE is fully specified without boundary ...
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30 views

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...
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Derivative of terminal state w.r.t. the inital conditions.

Let $x\in R^n$ and consider the system $$ \dot{x}=f(t,x) \;\;\mbox{with}\;\; x(0)=x_0 $$ and suppose that we know it's exact or very accurate solution $x(t)$ for the time interval $[0,T]$. I'm ...
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64 views

Please identify this equation: $\nabla^2 \mathbf F -k^2 \mathbf F = \mathbf A$

Is this equation $$ \nabla^2 \mathbf F -k^2 \mathbf F = \mathbf A $$ somehow named? F and A are vector fields. I guess inhomogeneous sign reversed Helmholtz equation isn't appropriate ...
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37 views

Stability properties of discretization of ODE

I am trying to find some conditions which guarantee that a continuous time dynamical system and it's discretization have the same behavior with regard to equillibrium points. Specifically that if the ...
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37 views

How to solve the following an ODE?

Let $x,y,z$ be a given point in $\mathbb{R}^3$. How to solve $(x'(t),y'(t),z'(t))=(x(x+y+z), y(x+y+z),z(x+y+z))$?
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126 views

Eigenvalues problem for generalized Kuramoto-Sivashinsky equation

I been working on Kuramoto-Sivashinsky Equation. In the process of analysis, I need to solve the following eigenvalues problem $$ -u_{xxxx}-\lambda u_{xx}=\beta(\lambda)u $$ where $\lambda$ is a ...
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ODE particular solution (physics)

I have to do this exercise: ($Z(t)=I(t)$, it's printed wrong). I have a doubt about the first item. To find all resonance when $R=1$, I found the particular solution $I_{p}(t)=A\sin(\omega ...
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For all topological conjugation $$h: \Delta_1 \rightarrow \Delta_2$$ we have to $h(\omega(p))=\omega(h(p))$, for all $p \in \Delta_1$

Let $X_1$ and $X_2$ fields in $\Delta_1,\Delta_2$ subset open in $\mathbb{R}^n$. Then, for all topological conjugation $$h: \Delta_1 \rightarrow \Delta_2$$ we have to $h(\omega(p))=\omega(h(p))$, for ...
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59 views

How to solve this complicated differential equation?

I need to know how to solve this complicated differential equation in $z$ either analytically or numerically : \begin{eqnarray} \frac{dx_1}{dz} &=& -ib_1x_1 - ikx_2 \\ \frac{dx_2}{dz} ...
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52 views

General solution of ODE

please what is the general solution of $$-(p(t)u')'+q(t)u=0$$ where $\displaystyle\frac{1}{p},\frac{1}{q}\in L^1((0,+\infty))$ Thank you
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72 views

Solution to Schrödinger equation $ \partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t)V(x)f(x,t).$

I want to solve $$ i\partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t-t_0)V(x)f(x,t),$$ for any $V \in C^{\infty}[-1,1]$ and $f: [-1,1] \times \mathbb{R_{\ge 0}} \rightarrow \mathbb{C}$. I would ...
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41 views

Power series to solve differential equations?

We can use the formula $$F(x)=e^{λx} [ ρ-λμ-\dfrac{1}{2} λ^2 σ^2 ]^{-1}. (1) $$ to derive an expression for F(x) when f(x) is any integer power $x^n$. Begin by observing that for the ...
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154 views

Is two-body motion planar?

The two-body problem studies the motion of two bodies under the influence of their gravitational attraction. Following the notation used in Wikipedia http://en.wikipedia.org/wiki/Two-body_problem, ...
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41 views

Differential Equations and Eigenvalues

I have the following system of differential equations: $$\left\{\begin{aligned} \frac {dx} {dt}=-4x+2y \\ \frac {dy} {dt}=-\frac 5 2x+2y \end{aligned} \right. $$ Which corresponds to the following ...