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

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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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220 views

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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337 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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277 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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323 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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135 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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149 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 ...
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87 views

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

The 'ring of differential operators wrt x' ? Thx.
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268 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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148 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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384 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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788 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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144 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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377 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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242 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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100 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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324 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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82 views

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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156 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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84 views

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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294 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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230 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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534 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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61 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 } ...
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50 views

What type of equation is this?

Is this equation an ODE or PDE $$ \frac{d^3u}{dx^3}−αxu=0, x∈R $$ The only thing given is $\int_R u(x) =\pi $ and $α>0$ is some constant. I have to find the solution using fourier ...
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32 views

Initial&Boundary Value problem-Fourier

$$u_{t}=u_{xx}, \hspace{5mm} x>0, t>0$$ $$u(0,t)=0 \hspace{3mm} u(x,0)=f(x)$$ We want that the solutions are bounded. We are looking for solutions of the form $$u(x,t)=X(x) \cdot T(t)$$ ...
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31 views

What would it mean if the solution to an IVP did not have maximal interval of existence?

I understand that it can be proved that for the IVP: $\frac{dx}{dt}=f(x)\quad$; $x(0)=x_0$ There is a maximal interval of existence $(\alpha,\beta)$, where there exists a unique solution. However I ...
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21 views

Partial Differential Equations Black Scholes Problem

Part 1) Consider the Black-Scholes problem $$\frac{\partial A}{\partial t}+\frac{\sigma^2B^2}{2}\frac{\partial^2A}{\partial B^2}+rB\frac{\partial A}{\partial B}-rA=0 ...
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ODE from systems biology, can I generalize this? Have solution but not sure how to arrive at it.

Reading a systems biology book, and it describes a model with the following ODE: $$ \frac{dY}{dt} = -\gamma Y + v_1 X_1 (T - Y) + v_2 X_2 (T - Y)$$ where $Y$, $T$, $T - Y$, $X_i$, $a$ and $v_i$ are ...
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Find the points at which an IVP admits at least two solutions

Given the IVP: $$\frac{dy}{dx} = x + |\sin(y)|$$ $$y(x_0) = y_0$$ Find the points in $\mathbb R^2$ at which this IVP admits at least two solutions. Clearly, $f(x,y) = x + |\sin(y)|$ is Lipschitz ...
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62 views

Solving second order nonhomogeneous differential equation with non-constant coefficients using Laplace Transform

$ty''(t) + y'(t) -ty(t)= tf(t)$ How to solve the problem using Laplace Transform? Using Laplace transform I got $$Y(s)= C(s^2-a^2)^{-1/2} + (s^2-a^2)^{-1/2}\int (s^2-a^2)^{-1/2}F(s)\,ds$$ where ...
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How can one prove the existence and uniqueness of solutions to linear differential equations?

It is a theorem (I think) that the equation: $$\mathbf{x}'(t) = A(t)\mathbf{x}(t) + \mathbf{b}(t); \qquad \qquad \mathbf{x}(t_0) = \mathbf{x}_0$$ Has a unique global solution for any matrix ...
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152 views

System of first order ODEs with coherent sinusoidal time varying coefficient

I have encountered equations of the form $$\frac{{d{\bf{y}}(t)}}{{dt}} = \left( {{A_0} + {A_1}\cos (\omega t)} \right){\bf{y}}(t)$$where ${\bf{y}}$ is a vector and ${{A_0}}$ and ${{A_1}}$ are square ...
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47 views

Differential Equation $(1+x^2)y'-2xy=\cot(x)$ after integrating factor

$$(1+x^2)y'-2xy=\cot(x)$$ or $$y'=\frac{2x}{1+x^2}y+\frac{\cot(x)}{1+x^2}$$ if I use an integrating factor $(e^{\int\frac{-2x}{1+x^2}dx}=\frac{1}{1+x^2})$ I get $$\frac{y}{(1+x^2)}=\int\frac{\cot ...
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30 views

Zeroes of a solution to a differential equation

Show that any solution to the equation $y''+xy=0$ has at least 15 zeroes on the interval $[-25,25]$. Please give me a hint.
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Is the following complex value differential equation always has a solution?

Let $a(z)$ be a fixed complex value complex variable function, not necessarily holomorphic. Consider the following differential equation $$ \frac{\overline{\partial}f}{\partial \overline{z}}+af=0. $$ ...
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Differential equation of the form to find

Lets $f(z)$ is some analytic function on complex plane and $y(z)$ is known analytic function on complex plane. Problem statement: find all $f(z)$ that: $$f(z) = f(z\frac{\partial}{\partial z})y(z)$$ ...
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25 views

Proving the Bessel function solves the Bessel equation

Using the notation for the Bessel function as $J_n(z)=\sum \limits_{k=0}^{\infty}\frac{(-1)^kz^{n+2k}}{k!(n+k)!2^{n+2k}}$, I want to show that $w=J_n(z)$ satisfies ...
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78 views

Periodic Solution of Damped Pendulum with Constant Torque

I have a system of ordinary differential equations $ \theta' = v$ $ v' = -bv - \sin \theta + k$ These are the equations for a pendulum with $\theta$ being angular position, and $v$ being angular ...
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79 views

Eigenvalues for $y''+2y'=\lambda y$

I must find the eigenvalues and eigenfunction for $$y''+2y'=\lambda y$$ with initial conditions $y(0)=0$, $y'(1)=0$. I have found the non-trivial case, and made an attempt to solve for $\lambda$, but ...
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18 views

Eigenvalue Function of Laplace Equation discretizes by nine-point stencil

I'm trying to plot the eigenvalue function of the Laplace equation $$-u_{xx}-u_{yy}=0,\;(x,y)\in (0,1)^2$$ with $$u(x,y)=0$$ on the boundary of the unit square. I have the nine-point stencil ...
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51 views

Using Green's Function to Find Particular Solution

We have the non-homogeneous differential equation $x^3y'''-3x^2y''+6xy'-6y=4x^2$ with conditions $y(1)=1, y'(1)=1, y''(1)=0$, and I have been tasked with finding its particular solution using Green's ...
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65 views

Show the IVP has a unique solution

Assume that $f:\mathbb{R}^{n}\times \mathbb{R} \to \mathbb{R}$ satisfies (i) there exists a constant $M>0$ such that $|f(x,t)-f(y,t)|\leq M|x-y|$ for each $x,y\in \mathbb{R}^n$ and each $t\in ...
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48 views

For time varying linear ODEs: is there a transformation which can make the system admit an exponential solution?

First, I'll start with the properties of the matrix in question: Assume we are given some matrix $\mathbf{A}(t)$, which is time dependent. This matrix is square and not invertible. A system: $$ ...
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Constructing a function using the Fourier transform

Pick an integer $n\ge 5$ and let $f\in C_{C}^{\infty}(\mathbb{R}^{N})$. We want to use the Fourier transform to formally construct a function $u\in L^{\infty}(R^{n})$ that solves ...
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Leading behaviour of DE at infinity

This is taken from the book of Bender and Orszag, problem 3.44. Find the leading behavior as $x\rightarrow+\infty$ of the differential equation: $x^3y'' - (2x^3 -x^2)y' +(x^3-x^2-1)y=0$ Explain ...
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What problems are related with the following type of FDE with delay?

Consider the following class of functional differential equations with delay: $$\begin{align} \frac{du}{dt} &= F(x,t,u(x,t),u_{t,x}), & (x,t) &\in [a,b] \times [0,T] \\ u(x,t) &= ...
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33 views

Why does an infinite Neumann boundary condition become a Dirichlet condition?

Often when I read a paper I see a statement of the type: Our boundary condition at the surface is $\frac{\partial f}{\partial x} = \alpha$. In the limit of $\alpha \to \infty$ this is equivalent ...
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28 views

Writing a 2nd order linear ODE as a set of 2 first order ODEs in its independent solutions

Given the homogeneous linear ODE $$y''(x)+ P(x) y'(x) + Q(x) y(x)=0$$ where $x\in(0,\infty)$ and $P$ and $Q$ are some smooth (but not necessarily bounded) functions. I know that we can write this as a ...