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

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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 ...
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80 views

Solving $c \sqrt{1+y'^2} - \rho g y + \lambda = 0$

I am trying to solve the following DE which corresponds to the hanging chain problem. Just need help to get started. Kinda stuck at that point. $\rho$, $\lambda$ and $g$ are all real. Thanks. $c ...
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134 views

ODE Show has unique solution.

I have this weird question from my book that I don't know how to solve. It is suppose that $p(x)$ is a Lipschitz function and $q(x)$ is continuous and then for the equation $x' = p(x)$ and $y' = ...
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23 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 ...
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83 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} ...
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52 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) + ...
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108 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) ...
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70 views

Question on finding solution to ODE

I have trouble solving the following ODE, I am wondering if I can get any hint or help The ODE I am facing is following $$\frac{\mu x}{y'(x)}=-A-x+\eta y(x)$$ where $A, \mu, \eta$ are all constant. ...
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57 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 ...
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53 views

Existence and Uniqueness of complex ODE's

I'm wondering if there is a theorem for the existence and uniqueness of complex ODE's. If there is, would someone mind explaining the general breadth of the theorem and/or directions to an online ...
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40 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 ...
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75 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 ...
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61 views

Linear equation solving

I cant solve $tx'+\dfrac{tx}{\sqrt{1+t^3}}=1$ I have tried to do it like an homogenian but i cant integrate $\dfrac{1}{\sqrt{1+t^3}}$ so i suposse it must be done by another method
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106 views

Help with simplifying Differential Equation

Original Problem: Solve Differential Equation $$\left(x^2 \sec{\left(y+1\right)} \tan (y+1)-3xe^{y+1}+\frac{2yx^{2}}{y^2+1}\right) \frac{dy}{dx}+3x \sec(y+1)-6e^{y+1}+3x \ln{\left(y^2+1\right)}=0$$ ...
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99 views

Solving ODE, with sinusoidal force and harmonic oscillator

OK, this is another one of those that should be simple enough to do. But maybe I am just bad at differential equations. We have the following: $$x''+ \gamma x'+ \omega_0^2 = 3 \cos t + 2 \sin t$$ ...
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26 views

How to Solve a Coupled Differential Equation? [duplicate]

I came across the set of following coupled equations while studying cycloid motion in Griffiths' Intro to ED $\ddot{y}=\omega \dot{z}$ $\ddot{z}=\omega (\frac{E}{B}-\dot{y})$ I am at a loss as to ...
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82 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) ...
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62 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} ...
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67 views

Question about a solution to a nonhomogeneous linear differential equation

(apologies for the format and somewhat vague title)
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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 $x(·)$ ...
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How to find Lie point symmetry?

I've been reading the Stephani's Differential equations: their solution using symmetries and I have a doubt in an excerpt. In the chapter 4, he is exposing the procedures on how to find Lie symmetry ...
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74 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 $\partial U^+ \cap \{x_n=0\}$. ...
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108 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} = ...
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483 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 ...
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1answer
153 views

solving bessel equation numerically.

Assuming there's an equation (bessel) and I'm told to solve numerically. This means, to solve this type of equation, we must convert the equation to a system of first order ODE's by letting $z=y'$ and ...
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203 views

Can we express all doubly periodic functions as one of doubly periodic function?

Singly Periodic Functions $e^{x},\cos(x),\sin(x),\tan(x), .. etc.$ Euler's identity is $$e^{i\alpha}=\cos(\alpha)+i\sin(\alpha)$$ $$e^{-i\alpha}=\cos(\alpha)-i\sin(\alpha)$$ Thus, we can express ...
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64 views

Solve a linear ordinary differential equation

Solve $ \dfrac{1}{x^2+1}\dfrac{dy}{dx} + xy = 3$, with $y(0) = 0$. Rearrange the equation I get $ \dfrac{dy}{dx} +(x^3+x)y = 3x^2 +3$ and the integrating factor is $$r(x) = e^{(\int x^3+x) dx} = ...
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149 views

Green' s function for harmonic oscillator

Does someone know how to get a solution of differential equation for Green's function $(-d^2/dt^2 + \omega^2) G(t, s) = \delta(t-s) $? There is a periodicity of G, actually $\Delta (t-s) = G(t,s)$ ...
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zeros of solutions of differential equation

Consider the equation $$\left(\frac{1}{1+t}x'\right)' + (1+ \sin{t})x = 0 $$ $a)$ Show that every solution of the equation has at least one zero in $ [0,\pi].$ $b)$ Show that there is a solution ...
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47 views

Problem with differential equation 3

I'm new with differential equation and I can't figure out how to solve this problem: The rate of change of temperature of an object is proportional the difference between the object temperature and ...
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28 views

How can I evaluate the time when the system get close to stable point without fully calculate the whole system

Consider the system $\begin{cases} \frac{dx}{dt}=f\left(x,y\right) \\ \frac{dy}{dt}=g\left(x,y\right) \\ \end{cases}$ ,with IC: $\begin{cases} x\left(t_0\right)=x_0 \\ y\left(t_0\right)=y_0 \\ ...
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102 views

How to determine if the differential is homogeneous or not?

I need help determining whether the following operators are linear homogeneous, linear inhomogeneous or non-linear and how can we know. This is is actually from my own knowledge, and I want to ...
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229 views

Characteristic functions in set theory

The book I am studying has the definition of a characteristic function as follows. Let $A\subseteq{X}$. Then $$\chi_A(x) = \begin{cases} 1, & \text{if $x\in{A}$} \\ 0, & \text{if ...
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571 views

Euler lagrange equation solving

Find the Euler-Lagrange equation for the functional $$I(y) = \int_0^1(py\,'\,^2-qy^2)\mathrm dx$$ subject to the constraint $$\int_0^1ry^2 = 1.$$ Answer: $\frac{d}{dx}(py') + (q-\lambda r)y = 0$. ...
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55 views

Conditions for transform second order PDE to a system of ODE?

What is the necessary conditions we need to transform second order PDE to a system of ODE? e.g. If I have $$a^2*u_{tt}- u_{xx}+ u*u_{x}=0 $$ what conditions I needed to transform it to a system of ...
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Is the one demential time-independent Schrödinger equation solvable in potential (1+Tanh(x+1))(-1+Tanh(x-1))

The one dimensional time-independent Schrödinger equation reads: \begin{equation} -\frac{h^2}{2m}\frac{d^2\psi}{dx^2}+U(x)~\psi=E~\psi \end{equation} where $\psi(x)$ is the wavefunction, U(x) is the ...
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107 views

Finding a lipschitz constant for an interval in a system of ODEs

I'm trying to approach a question and my grasp on Lipschitz is a little rusty. My approach will be to take the max norm of the jacobian for my system and to set that as my constant. My ODE is $\ ...
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63 views

A BVP question using green's function

When doing exercise, I found this question with boundary conditions I couldn't solve. y′′ + y = f(x), 0 < x < 2π, y(0) − y(2π) = 0, y′(0) − y′(2π) = 0 The question is asking what goes wrong in ...
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70 views

Modeling with Linear Differential Equations

Working through a homework problem for my differential equations course. We're modeling the time it takes to cure a staph infection based on a dosage of antibiotic. We're given the initial value ...
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90 views

Second order linear homogeneous ODE with constant coefficients and repeated roots. Why second solution needed?

In this case of a second order linear homogeneous ODE with constant coefficients and repeated roots: $ay'' + by' + cy = 0 $ (and $r1 = r2$) why is the solution $y_1(t) = e^{-b/2a}$ not enough? in ...
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Checking on PDE problem, done correctly?

I'd have posted the equation itself in the headline but this seemed simpler. OK, a PDE: $$a\frac{\partial u}{\partial t} + b \frac{\partial u}{\partial x} = u$$ So, I'll set up a couple of ...
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53 views

Can the system $\partial_x f(x,y) = \dot{y}$, $\partial_y f(x,y) = \dot{x}$ be related to some Hamiltonian system?

If one has found some function $f(x,y): \partial_x f = \dot{y}, \partial_y f = \dot{x}$, is there a simple transformation or change of variables that results in Hamilton's equations $\partial_p H = ...
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284 views

Solution of Bessel equation

Prove that for a Bessel function in its normal form that is: $$u'' + \left(1 + \frac{1-(4*p^2)}{4x^2}\right)u=0$$ if $p > \frac12$ then every interval of length $\pi$ contains at most one zero of ...
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174 views

Wrong answer for this differential equation temperature problem.

(a) An object is placed in a 68°F room. Write a differential equation for H, the temperature of the object at time t. ANSWER: dH/dt = -k(68 - H) (b) Give the general solution for the differential ...
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76 views

Convergence of solutions in initial value problem

I am working on the following problem: Suppose $u_{n} : [-M, M] \rightarrow \mathbb{R}$ are differentiable and are such that $u_{n}'(x) = F(u_{n}(x), x)$ for $F$ continuous and bounded. Furthermore, ...
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42 views

Binet's differential equation yields a circular path

Binet's equation is $$\frac{d^2u}{d\theta^2}+(1-\frac{\lambda}{a^2v^2})u=0$$ I need to show that for $\lambda=a^2v^2$ path will be a circle. For $\lambda=a^2v^2$ I have $$\frac{d^2u}{d\theta^2}=0$$ ...
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34 views

Issues in calculating the gradient

I am trying to calculate the gradient of a certain expression. I am not sure if it's possible. I have the following $f(\alpha_1,\alpha_2,\Lambda) = \log(|2Q_1+2Q_2 +2Q_3|)$ $Q_1$ is a diagonal ...
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60 views

Solving ODE with negative expansion power series [duplicate]

I am solving a series of ODE, such that each DE is equal to some degree of term that I'm expanding to. For instance, one DE is this: $\xi^r\partial_r g_{rr}+2g_{tt}\partial_t\xi^t=\mathcal{O}(r)$ ...
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67 views

partial differential equation problem

I have the pde $$\ \dfrac {\partial u}{\partial t} + \ 3\dfrac {\partial u}{\partial x}=0.$$ $$ x>0, t>0$$ $$u(x,0)=e^{-x^2}, x>0,$$ $$ u(0,t) = e^{-t^2}, t>0 $$ The characteristics ...
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2answers
86 views

stability of a linear system

The linear system: $y''(t)+4y'(t)=4(\lambda -1)y(t)+z(t)$ $z'(t)=(\lambda -3)z(t)$ Determine the stability of the system as a function of the parameter $\lambda\in\mathbb{R}$. ...