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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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Separable equations and implicit differentiation and differential form.

I'm a tad confused about some of the symbolic notation in my book regarding separable equations: I thought when you integrate an equation (let's call it a) and differentiate it, you wind up with a ...
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1answer
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How is stability for a numerical solution generalized to a system of ODEs?

Consider the system of ODEs $$y' = \begin{bmatrix}-6&4\\4&-6\end{bmatrix}y, \quad t\in[t_0, t_e], \quad y(t_0)=y_0$$ I'm asked for what stepsize the explicit Euler method generates a stable ...
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Fokker planck equation, simple change of variables

I’m studying this fokker planck equation $\frac{\partial}{\partial t} p + a \frac{\partial}{\partial x} p = D \frac{\partial^2}{\partial x ^2}p $ where p is a fuction of x and t. a, D constant. Now ...
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Stability region of explicit midpoint method [duplicate]

Consider the explicit midpoint method, i.e $$y_{n+1}-y_{n-1} = 2hf(y_n).$$ I'm asked to apply this method to the linear test equation, $f(y_n) = \lambda y_n,$ in order to find the method's stability ...
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variation of parameters 1st order ODE Version

I have been given a variation of parameters formula for a first order ODE $x'(t) = a(t)x(t)+b(t)$ and have been asked to differentiate it, the formula is: $$ x(t) = Ce^{\int_{t_0}^{t}a(s)\,ds} ...
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help - A system consists of springs A and B and objects C and D

Hi all, I have completed the question up to part c). Using matlab, I have found the eigenvectors/eigenvalues but from there I am unsure as to how to find the solution which would comprise of the below ...
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Can we cancel the equality mark here?

Problem Let $f(x)$ satisfy that $f(1)=1$ and $f'(x)=\dfrac{1}{x^2+f^2(x)}$. Prove that $\lim\limits_{x \to +\infty}f(x)$ exists and is less than $1+\dfrac{\pi}{4}.$ Proof Since $f'(x)=\dfrac{1}{x^2+...
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Find number of functions $f$ which satisfy given conditions

Let $\ln (x)$ denote the logarithm of $x$ with respect to the base $e$. Let $S ⊂ \mathbb R$ be the set of all points where the function $\ln(x^2 − 1)$ is well defined. Then the number of ...
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Construct a differential equation whose solution in parametric form is the butterfly curve.

Is it possible, and if so, does anyone know how to construct a differential equation whose solution on parametric form is the butterfly curve: $$x=\sin (t)\left(e^{\cos (t)}-2 \cos (4 t)-\sin ^{5}\...
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2answers
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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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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
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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
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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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2answers
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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
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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
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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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1answer
27 views

ODE with discontinuous vector field

Consider the ODE $$\partial_t \Phi(t,x) = \mathbf b(\Phi(t,x)), \qquad t \in [0,T], \quad x=(x_1,x_2) \in \mathbb{R}^2$$ $$\Phi(0,x) = x, \quad x \in \mathbb R^2,$$ where $\mathbf b = (0,\chi_{\{x_1 \...
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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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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
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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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1answer
37 views

Using a Lyapunov function to determine stability of equilibria

Given $$\left\{\begin{aligned} x' &= -x^3 + 7xy^2\\ y' &= -3x^2y+y^3\end{aligned}\right.$$ find $a, b > 0$ such that $L(x,y) = a x^2 + b y^2$ obeys $\frac{d}{dt}L \neq 0$ whenever ...
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1answer
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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
42 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
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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
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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
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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
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How do i solve this differential equation. [on hold]

I'm not really concerned with the answer, im just wondering how the last two lines are equal.
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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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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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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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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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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
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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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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
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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
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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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1answer
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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
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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
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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
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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
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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
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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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1answer
29 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? Someone ...
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1answer
17 views

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
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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-{\...