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Questions tagged [ordinary-differential-equations]

For questions about ordinary differential equations, which are differential equations containing only derivatives w.r.t. one variable. For questions specifically concerning partial differential equations, use the [tag:pde] instead.

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Finding solution to a non-linear differential equation

I am given the equation below, where b is the average number of births (b=8), and d is the average number of deaths (d=3), and I am given an initial condition: P(0) = 500. Given the above values, I ...
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5 views

Why does the rotating wave approximation work?

Consider two coupled oscillators with position coordinates $X_a$ and $X_b$. In general, the motion is described by a system of coupled first order linear differential equations: $$ \frac{d}{dt} \begin{...
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1answer
22 views

Paul's Online notes example seems to be wrong? Variations of parameters diff equation

Paul's Online Note Example As far as I know, the complementary solution only consists of cos and sin when the roots are complex. From what I see, the roots for this equation are r=0 or r=-9. Not ...
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1answer
11 views

Differential equations error of magnitude question

Let $x = x(t), y = y(t)$ be the solution to the initial-value problem $$\frac{dx}{dt} = -x - y, \hspace{1em} \frac{dy}{dt} = 2x - y, \hspace{1em} x(0)=y(0)=1.$$ Suppose that we make an error of ...
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43 views

How to remove this numerical artifact?

I am trying to solve a differential equation: $$\frac{d f}{d\theta} = \frac{1}{c}(\text{max}(\sin\theta, 0) - f^4)~,$$ subject to periodic boundary condition, whic would imply $f(0)=f(2\pi)$ and $f'(0)...
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1answer
30 views

How to prove $\lim_{n\to\infty}\sup A(t)\le\frac{a}{b}$?

Suppose $A(t)>0(t\ge 0)$, $a, b>0$, let $$ A'(t)\le aA-bA^2. $$ Prove $\lim_{n\to\infty}\sup A(t)\le\frac{a}{b}$. Using Taylor formula $$ A(0)=A(t)-tA'(t)+o(t)\ge (1-ta)A(t) +tbA^2(t)+o(t). ...
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1answer
65 views

Proving that any solution to the differential equation of an oscillator can be written as a sum of sinusoids.

Suppose you have a differential equation with n distinct functions of $t$ where $\frac{d^2x_1}{dt^2}=k_{11}x_1+...k_{1n}x_n$ . . . $\frac{d^2x_n}{dt^2}=k_{n1}x_1+...k_{nn}x_n$ I want to show ...
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0answers
12 views

ODE with discontinuous vector field

Consider the ODE $$\partial_t \Phi(t,x) = \mathbf b(\Phi(t,x))= 0, \qquad t \in [0,T], \quad x=(x_1,x_2) \in \mathbb{R}^2$$ $$\Phi(0,x) = x, \quad x \in \mathbb R^2,$$ where $\mathbf b = (0,\chi_{\{...
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Solving a differential equation with an unknown relation

Say we have an unknown function $x(y)$. But we do know that $f(x) + g(y) = {{dx} \over {dy}}$, where $f(x)$ and $g(y)$ are known. Is it possible to find $x(y)$ like this? If not, what can we know ...
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2answers
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Solving $\frac{dy}{dx}=\sqrt{3x+2y}-\frac{3}{2}$ without stuff from higher-order differential equations

I'm trying to solve this equation: $$\frac{dy}{dx}=\sqrt{3x+2y}-\frac{3}{2}$$ without using stuff from higher-order differential equations. I've tried using substitution $ w=\frac{y}{x} $, but ...
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0answers
9 views

Uniqueness of integral curves starting on boundary?

I'm only aware of ODE uniqueness on open subsets of $R^n$, which is useful when we want to prove uniqueness of integral curves on a smooth manifold. Suppose we have an interal curve $f:[0,\delta]\to ...
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Prove that $f(x,0)=f(0,x)$ for all X (Hint:Use A1) Let f(x):{ f(x,0) if x≠0, Let f(x):{ 1 if x=0 [on hold]

Consider throwing a dart at the origin of the Cartesian plane. You are aiming at the origin, but random errors in your throw will producr varying results. We assume that: 1.) The errors do not depend ...
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1answer
26 views

Picture flow of ODE

Consider the ODE $$\begin{cases} \frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0, \quad x \in \mathbb R^2 \\ \Phi(x,0) = x, \quad x \in \mathbb{R}^2 \end{cases}$$ Suppose that the flow $\Phi$ ...
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0answers
22 views

Using a Lyapunov function to determine stability of equilibria

For each of the following systems, find $a>0$ and $b>0$ such that $L(x,y)=ax^2+by^2$ obeys $\frac{d}{dt}L\neq 0$ whenever $(x,y) \neq (0,0)$. State whether the origin is a stable or unstable ...
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1answer
15 views

What is the general solution of $xy=(x^2+4y)(dy/dx)?$

I have tried to use u-substitution to separate the variables, as well as distributing $dy/dx$ as y' to $(x^2+4y)$. However, I just can't seem to separate the variables no matter what methods I've ...
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1answer
41 views

Series solution about $x=0$ of $xy''-y'+4xy=0$.

I want to find at least one solution of the differential equation $$xy''-y'+4xy=0$$ about the point $x=0$. I identified that $x=0$ is a regular singular point and thus Frobenius Theorem is applicable. ...
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2answers
21 views

Long term behaviour of trig functions

In answering a differential equation question, the question asked me to solve the equation and then give the function to which $y$ approximates when $x$ is large and positive. To which I have no idea ...
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1answer
39 views

How to prove that the solution of $\frac{dy}{dx}=1+y^4$ with $y(x_0)=y_0$ cannot be extended to $\infty$ and $-\infty$

A friend of mine gave me a proof, but I didn't get it. If the solution can be extended to +∞ $$ \frac{dy}{1+y^4} = dx $$ Integrate $x$ from $x_0$ to +∞ So $$+∞ = \int_{x_0}^\infty \frac{ dy(x)}{1+...
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1answer
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$3y''+4y'+y = 200 \cos x$ finding particular integral

$$3x''+4x'+x=200 \cos(t) $$ This is what I did: \begin{align} x &= \lambda \cos (t) + \mu \sin(t) \\ x' &= -\lambda \sin(t) + \mu \cos(t) \\ x''&= -\lambda \cos(t) - \mu \sin(t) \end{...
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1answer
58 views

How do you solve this kind of third order ODE? [on hold]

For this given equation: $y'''(x)+y(x)y''(x)=0$ How do you get the solution for this ODE?
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1answer
34 views

How do i solve this differential equation.

I'm not really concerned with the answer, im just wondering how the last two lines are equal.
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2answers
16 views

variable co-efficient 2nd order linear ODE

I am trying to solve a variable co-efficient 2nd order linear ODE by using a transformation for the independent variable: $y'' + \frac{2}{4x} y' + \frac{9}{4x} y = 0$ with transformation $t = \sqrt{x}...
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0answers
28 views

Initial conditions of ODEs: Why is $\frac{𝑑𝑢}{𝑑𝜙} = \frac{1}{𝐷 \cdot 𝑡𝑎𝑛(𝛼)}$?

I've started reading this paper (kindly check out the paper for more context, figure 1 provides a visualization of the problem) where it describes light trajectories on a Schwarzschild spacetime and ...
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0answers
33 views

Existence of a periodic solution?

The sign of the result will change only depending on how $y$ changes. I need it to always be negative or always be positive. Is that possible here? I'm looking for the existence or nonexistence of a ...
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0answers
30 views

Using collocation method to solve a nonlinear boundary value ODE

I have the following ODE $$ u'' = -(1 + e^u), \quad u(0)=0,\quad u(1)=1$$ I want the divide the interval $[0,1]$ into $n-1$ equal subintervals each with length $h=1/(n-1)$ and we take approximate ...
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91 views

2nd Order ODE involving radial velocity

Question: A rigid disk is rotating with a constant angular velocity of ω. There is a smooth groove on the disk that allows free motion of a particle along the radial direction of the disk. At t=0, a ...
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1answer
28 views

Reduction of order method for second order linear ODE

Given the following differential equation $$x^2 y'' + x(2x^2 + 1)y' + (2x^2 -1)y =0$$ solve it using the reduction of order method. The given solution is $$y_1(x) = \frac 1x.$$ I have been ...
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0answers
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Bendixson negative test or an ellips

Apparantly, this system of equations doesn't have a periodic solution: $$x' = x(1-x^2-3y^2)$$ $$y' = y(3-x^2-3y^2)$$ We used in the Bendixson negative test in class so $$f_x = 1-3x^2-3y^2$$ $$f_y ...
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1answer
33 views

Using finite differences to solve BVP

I have the following ODE $$ u'' = -(1 + e^u), u(0)=0, u(1)=1$$ Using a second order accurate finite difference I obtain $$ -(1+e^{u_i}) \approx \frac{ u_{i+1} - 2 u_i + u_{i-1} }{h^2} $$ and $...
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1answer
41 views

Bessel differential equation from integral

It is a relatively well-known fact that $$\int_{0}^{2\pi}e^{-ikr\cos\theta}d\theta=2\pi J_{0}(kr),$$ where $J_{0}$ is the Bessel function of the first kind and order zero. I'm trying to show that this ...
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2answers
29 views

Differential Equation by Minimization

Suppose we want to solve $u + xu' = 0$, which has the general solution $u = \frac{C}{x}$, by minimizing the length squared of $u + xu'$. This should work due to the positive definite condition of ...
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1answer
34 views

Trying to solve a two-point boundary value problem on MATLAB

I have the following ODE $$ u'' = -(1 + e^u), u(0)=0, u(1)=1$$ with boundary conditions and $t \in (0,1)$. I want to solve this ODE using the shooting method. First,we convert this to a system of ...
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1answer
31 views

Phase Portrait vs Explicit Solution

Currently revising Ordinary Differential equations and I seem to have come across a contradiction Now my confusion comes with the sign of $x_0$ surely using the phase portraits if $x_0 < k$ when $...
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1answer
29 views

About interval of definition for solution to an ODE

Let $f:\mathbb{R}\times\mathbb{R}^{n}$ continuous such that $x'=f(t,x)$ has uniqueness of solution, and $|f(t,x)|\leq 10$. I wnat to prove that every solution for the ODE is defined for all $t\in\...
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1answer
28 views

On ODE with uniqueness solution

Let $f:\mathbb{R}\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ continuous and Lipschitz. Let $\gamma(t)$ solution for the Cauchy Problem: $$ \begin{cases} (x,y)'=f(t,(x,y)) \\ (x,y)(0)=(7,-10) \end{...
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1answer
27 views

Simultaneously bounding stable and unstable components

I am reading a passage from Perko's book about the Stable Manifold Theorem. Here is the problem: Let $\dot x = f(x)$ be a system where $f: E \subset \mathbb{R}^{n} \rightarrow \mathbb{R}^n$ (with $E$...
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About the relationship between $f,g$ and its solutions for ODE

This is a true or false question. Let $f(x,t)$ continuous and $\dot{x}=f(t,x)$ an ODE with unicity of solution. So, given $\epsilon >0$, exists $\delta >0$ such that $$|g(x,t)-f(x,t)|<\...
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Discuss the existence or nonexistence of a periodic solution

I am solving a question where I need to discuss the existence or nonexistence of a periodic soltuion and find the region in the plane where my results holds. $$x'' + \Bigr(3-(x')^2\Bigr)x' + x = 0$$ ...
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19 views

Differential Equations- Reduction of order

Why is the equation in the red rectangle true? Why is it that if I have one solution y1(x), the second solution can be written as y2(x)=v(x)*y1(x)?Is it because they are linearly independent?
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1answer
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Derivative of eigenvalue with respect to a constant

I am having trouble wrapping my mind against a simple problem: Suppose we have the following eigenvector equation for $A\in\mathbb{R}^{n\times n}$ and $\alpha \in \mathbb{R}$. $$ \left(\alpha A\right)...
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1answer
32 views

Linear ODE Cauchy problem: issue in the distributional solution

I am considering the following Initial Value problem: $$ \begin{aligned} &u'+\alpha u = \cos \omega t\\ & u(0)=u_0 \end{aligned} $$ The solution is: $$u(t) ={\rm e}^{-\alpha t} \left(u_0-{\...
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2answers
56 views

Need help with this differential equation $\frac{dh}{dt}=\frac{5}{h^2}-\frac{1}{20}$

I am an A level student and am stuck on this differential equation. I know it is simple for most of the brilliant minds here but I have been trying for an hour with no good result. $$\frac{dh}{dt}=\...
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34 views

Writing out a differential equation

Someone deposits money in a bank account at a continuous rate of $5000$ per year, and the account earns interest at a continuous rate of $7\%$ per year. Is the differential equation for the balance $B$...
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Phase, Isochrons, Isochrons map and Lift

at the moment i read the following paper: https://arxiv.org/pdf/1512.04436v1.pdf I have some questions about it and i hope someone can help me. On page 4/5 they introduce isochrons and the isochron ...
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66 views

Poincaré-Bendixson Theorem

It can be shown that $D = \{(u,v): u \geq 1/16, 0 \leq v \leq 128, u+v\leq 130\}$ is an invariant region for a system of differential equations based on a trimolecular reaction model $$u' = (1/8) - u ...
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73 views

Test for the existence or nonexistence of a periodic solution

Discuss the existence or nonexistence of a periodic solution for the following equation. Find the region where your result holds. $$x'' + \big(3-(x')^2\big)x' + x = 0 $$ At first glance, I thought ...
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16 views

Power series solution for nonlinear ODE

I am struggling to find a series solution for the initial value problem $y’=2y^2-3y+2$, $y(0)=y_0$. By plugging in a power series, I have simplified and gotten $$\sum_{n\ge0}[(n+1)a_{n+1}-2(a*a)_n+...
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0answers
22 views

Relationship between numerical integration and differential equation solving

everyone! I am starting an internship in numerical analysis, and I was hoping someone could help clarify a slight disconnect I have in how we find numerical solutions to differential equations. When ...
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0answers
15 views

How to tell when a state variable violates a bound during a numerical ODE solve?

Is there a good way to tell when a state variable violates a bound during a numerical ODE solve? For example, say we're simulating an object flying through the air with an ODE solve, it'd be nice to ...
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1answer
23 views

Plot solutions of ODE with discontinuous source

Consider the ODE $$ \begin{cases} \partial_t \Phi(t,x) = H(\Phi(t,x)), & t>0, x \in \mathbb{R} \\ \Phi(0,x) = x, & x \in \mathbb R, \end{cases} $$ where $H$ is the heaviside function. How ...