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

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Find functions that satisfy a given differential relation

If I have a relation between two sets of functions $A_{i}(x,y,...,z)$ and $B_{k}(x,y,...,z)$ of the form $$ A_i = F_i(B_k, \partial B_m/\partial x_n) \tag{1} $$ that is: $A$s are functions of ...
2
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0answers
46 views

Recurrence equation approximation

I have the following recurrence relation, $$x_{i+1}=a\cdot x_i^{\frac{2-2\alpha}{3}}+x_i,$$ where $a>0, \alpha>0$, and $x_0>0$. My goal is to get an approximate the expression for $x_i$. I ...
0
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0answers
6 views

Radial trajectory equation solution (large trajectories/$g$ is not constant)

Well for a simple radial trajectory one could create the following equation ($s$ being the distance from the origin $G$ being newton's gravitational constant, and $m$ the mass): $$\ddot{s} = ...
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0answers
14 views

dominant balance for coupled differential equations

I have been trying to solve following set of nonlinear differential equations: $\frac{dy(t)}{dt}=k z(t) - 3 k y(t) - y(t)^2 + \epsilon_1 (M-z(t))^2$ $\epsilon_2 \frac{dz(t)}{dt}=Mz(t) - z(t) y(t) - ...
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0answers
16 views

solve the question by charpit's method please [closed]

i need this question to be solved by charpit's method, it is urgently needed as i am doing my preparations for the annual exam. i shall be thankful. px+qy=y
1
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1answer
35 views

$y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$

Let , $y(x)$ be a continuous solution of the initial value problem $y'+2y=f(x)$ , $y(0)=0$ , where, $$f(x)=\begin{cases}1 & \text{ if } 0\le x\le 1\\0 & \text{ if } ...
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0answers
10 views

strong minima and maxima condition in calculus of variation

I am going through the topic CALCULUS OF VARIATION. There are not many examples on the topic strong/weak maxima minima. Can anybody provide the link of the source or book name where this topic is ...
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0answers
29 views

Uniqueness of the solution of a PDE system

If I have the following PDE system: $\frac{\delta}{\delta t}x(t,r)=-\int_0^1 G(|r-r'|)y(t,r')dr'x(t,r)$ $\frac{\delta}{\delta t}y(t,r)=\int_0^1 G(|r-r'|)y(t,r')dr'x(t,r)-y(t,r)$ $x(0,r)=a(r), ...
2
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1answer
31 views

Function of the trajectory of a differential equation

I want to show that there is no continuously differentiable non-constant function $H : \mathbb{R}^2 \to \mathbb{R}$ with $\nabla H(x,y) \neq (0,0)$ so that for every solution of the differential ...
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1answer
24 views

Prove that each $W_0^{1,2}$-function is weakly differentiable

Let $\Omega\subseteq\mathbb R^n$ be open. $u\in\mathcal L^1_\text{loc}(\Omega)$ is called weakly differentiable $:\Leftrightarrow$ $\exists v\in\mathcal L^1_\text{loc}(\Omega;\mathbb R^n)$ with ...
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1answer
25 views

Differential equation of waves

The differential equation of the spring mass system gives you a second order differential equation. Now, similarly wave equations have solutions that just like the spring system contain trigonometric ...
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0answers
43 views

hamiltonian calculations [closed]

Can somebody help me with this current value hamiltonian question please, will have an exam on similar style questions; Minimize $$ \int_0^2 (x^2 + u^2)e^{-0.03t}dt, \qquad (x = x(t),u=u(t)) $$ ...
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0answers
22 views

Current value Hamiltonian questions [closed]

Would really appreciate it if somebody could solve this question, we will have similar ones on the next math exam and I would love to know how to answer it. Minimize $$ \int_0^2 (x^2 + ...
5
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0answers
51 views

Differential equation $f'(x)=\alpha\cdot f(x-1)^\beta$

Is there a way to solve $$f'(x)=\alpha\cdot f(x-1)^\beta,$$ where $\alpha>0,\beta\neq0.$ I know that if the arguments matched, I could use separation of variables to get ...
1
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0answers
15 views

Reference needed for a short time existence result of quasilinear PDE on a compact manifold (relating to Ricci flow).

I'm currently in the proces of learning and writing a bit about the Ricci flow. In particular I'm studying the case of compact 2d Riemannian manifolds. Mostly I'm making good progress but I do miss ...
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0answers
20 views

uniqueness for complex differential equation.

I have a problem understanding an exercise regarding kubo martin schwinger boundary conditions from a text book on nonequilibrium greens functions. I have the following complex differential equation ...
1
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1answer
38 views

exact solution to lotka-volterra equations

I am looking for exact or perturbative solution realistic lotka-volterra (the one with logistic term in one of the equations) equations in population dynamics. Any reference where they have done it ...
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4answers
84 views

How to solve $\frac{d^{2}y}{dx^2} + \frac{1}{x} \frac{dy }{dx} =0$? [closed]

Let $y\in C^{2}(\mathbb R)$ (twice continuously differntiable function). We consider the ODE as follows: $$\frac{d^{2}y}{dx^2} + \frac{1}{x} \frac{dy }{dx} =0$$ ($y$ is function of $x$) My naive ...
2
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3answers
47 views

Undamped Pendulum Phase Plane Solution

Given the following ODE which is supposed to represent an undamped pendulum, with x representing the vertical angle: $$\frac{d^2x}{dt^2}= -2\sin(2x)$$ Make the substitution $$\frac{dx}{dt}= y $$ ...
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1answer
17 views

Problem Involving Eigen Functions/Values in Differential Equation

I am confused about finding eigen values/functions for the following exercise. $$y'' - \lambda y = 0 , y(0) = 0, y'(L) = 0 $$ When $$ \lambda =0 $$ I find that $$ y = c_1cos(x) + c_2sin(x) $$ $$ y' ...
2
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1answer
30 views

How do I demonstrate that the given functions solve this system of ODEs?

The system is $$\left\{ \begin{array}{rcl} x'&=&y-x(x^2+y^2-1) \\ y'&=&-x-y(x^2+y^2-1), \end{array} \right.$$ and the given solution is $$x(t)=\sin(t), \quad y(t)=\cos(t) .$$ ...
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1answer
27 views

Finding a second solution to a 2nd order differential equation.

So for these problems it asks us to use reduction of order to find a second solution to the differential equations. My professor said that there exist other methods to solve these problems. I am ...
2
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2answers
226 views

Find the derivative of the inverse of this real function $f(x) = 2x + \cos(x)$

I don't know how to attack this problem. The last I've tried is using a differential equation, but I don't know how to solve it. Let $y$ be $f^{-1}(x)$. Knowing that $x=f(y)= 2y + \cos(y)$ and ...
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0answers
25 views

Analytic Solutions To Matrix Differential Equation

Given the matrix differential equation: $\frac{d U(t)}{dt} = A(t) U(t)$ and the fact that $A_t$ is comprised only of analytic functions Is it possible to conclude that the solution $U(t)$ will be ...
2
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0answers
84 views

Validity of approximating a difference equation with a differential equation

Consider the following two equations: $$ \begin{cases} A_k(n+1)-A_k(n)=\beta \displaystyle \frac{ A_{k-1}(n)- A_k(n)}{\alpha+ n} + \delta_{k, \beta} \\ \\ \displaystyle ...
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4answers
85 views

Difference between $f(x(t))$ and $f(t,x)$

Why is there a difference between the two differential equations: $\overset{.}{x}(t)=f(x(t))$ and $\overset{.}{x}(t)=f(t,x)$ ?
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2answers
41 views

what can you say about the solutions of the equation $y' = x^2+y^2$ just by looking at the differential equation

Can we say that the graph is symmetric about origin. Because replacing $x$, $y$ with $-x$, $-y$ does not change the equation Also the slope becomes larger as we move away from origin. Anything else ...
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1answer
35 views

Does any differentiable function admit an expansion $\sum^{\infty}_{n=0}a_{n}x^{n}$?

Let $f(x)$ be a differentiable function in some interval $D$. Then does that mean that we could always write f(x) in the form $\sum^{\infty}_{n=0}a_{n}x^{n}$ in that interval?
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2answers
12 views

Ordinary Differential Equations - Harmonic Oscillator Question

A 3kg mass is attached to a spring with spring coefficient k = 48N/m. The mass is initially 0.5m to the left of equilibrium and at rest when it is let go. If the friction is negligible, find the ...
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0answers
46 views

When is one or more PDEs equivalent to one or more ODEs?

I'm relatively new to PDEs and ODEs. It seems that PDEs are generally more difficult to solve than ODEs, and so I intuitively have the feeling that one needs more information/knowledge/theorems in ...
2
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2answers
46 views

Physical applications of Chebyshev's equation.

As reported by Wikipedia, Chebyshev's equation is the second order linear differential equation $$(1-x^2) {d^2 y \over d x^2} - x {d y \over d x} + p^2 y = 0 $$ where $p$ is a real constant. Has ...
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2answers
41 views

Solution of Second order ODE: theoretical question

I know that the way to solve a second order ODE is to find a solution that has the form $y=e^{\lambda x}$ (solution for the homogeneous equation), but I haven't understood if solutions that aren't ...
0
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1answer
40 views

Calculate multivariate Gaussian from univariate Gaussian

I am currently trying to solve an exercise that involves estimating the position $\chi_t$ and and velocity $\dot\chi_t$ of a truck at time $t$. The truck moves on rails and is buffeted around by a ...
0
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1answer
32 views

How can I relate two systems of ODEs, when their initial conditions are related?

Consider two simple ODEs with identical right-hand sides, whose initial conditions are related by a simple formula. $$x' = a \cdot x, \quad y' = a \cdot y, \quad x(0) = \alpha \cdot y (0)$$ The ...
3
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1answer
98 views

Convergence of Integral of Matrices

I have proved the following convergence, but I'm not convinced with my answer. Suppose we have the following $$\lim_{t \rightarrow \infty} \text{tr}\int_{0}^{t}e^{sQ^T}Pe^{sQ} ds$$ where tr is trace. ...
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1answer
28 views

Equilibrium Points for 8th Degree Polynomial

I have an 8th degree polynomial that I need the zeros for. Is there even a way to explicitly solve one? Its for a diff equations review. I need to sketch the phase line, which is a breeze once I get ...
13
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3answers
494 views

Fourth Order Nonlinear ODE

I was looking at an ode $w^{(4)} + w^3 = 0$ with initial conditions $[w'''(0),w''(0),w'(0),w(0)]=[1,0,0,0]$. I can see via maple that there is a blowup around 3.7. I was wondering if there was a way ...
2
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1answer
31 views

Unique solution of Volterra integral equation of second kind

Dear Maths Stackexchange, In the context of a physics problem, I am looking at a linear integral equation, more specifically a 2nd kind Volterra equation in the unknown $g(t)$: \begin{equation} ...
4
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1answer
50 views

Completeness of the vector field $e^{-x} \frac{\partial}{\partial x} + \frac{\partial}{\partial y}$

I just want to bounce this off of the smart people on MSE to make sure I understand what's going on when we discuss complete vector fields. Consider the following field. $X = e^{-x} ...
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1answer
64 views

A differential equation I

Consider the second order differential equation \begin{align} 2 t^{3} y'' + (5 t^{2} - t) y' + (t^{2} - t + 1) y = 0 \end{align} with the conditions $y(0) = 0$ and $y'(0) = 1$. A solution is known in ...
3
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2answers
38 views

Solution of Second Order Differential Equation with non-constant coeffecient

How do we solve the differential equation $y''-2(\sin x)y'-(\cos x-\sin^2x)y=0$ IVP: $y\left(\dfrac{\pi}{2}\right)=0$ , $ y'\left(\dfrac{\pi}{2}\right)=1$ ? Its neither constant coeffecient, nor ...
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0answers
14 views

Reciprocal relations in Roulette /glissette rollings

If a catenary rolls on a straight line its focus traces out a parabola and vice versa. Is it true? Are there more such examples and how are they co-related? In case of a circle rolling on a fixed ...
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1answer
97 views

Kovacic's algorithm

Is there any reference with some example, about how to solve a "riccati" equation in this (below) form :$$y'(x)+a(x)y^2(x)+b(x)y(x)+c(x)=0$$ by Kovacic's algorithm? Or can anybody help me to ...
4
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1answer
58 views

General closed form solution to $f'(x) = P(f(x))/P(x)$

Does there exist a general closed form solution (in terms of elementary or special functions) to the differential equation: $$ \frac{df(x)}{dx} = \frac{P(f(x))}{P(x)} $$ when $P(x)$ is a polynomial ...
3
votes
1answer
41 views

Derivative of solution of differential equation with respect to parameter

Find derivative with respect to $A$ of solution of differential equation $$ \ddot x = {\dot x}^2 + x^3\tag1\label1 $$ with initial conditions $\{x(0)=0,\,\dot x(0)=A \}$ at $A=0$. My attempt We can ...
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2answers
356 views

why the standard deviation is not as the same as online calculator

I need to calculate the standard deviation for these numbrs: -12 -3 0 -13 8 -6 0 -22 -1 7 -7 1 -2 -13 -4 0 -6 -4 -10 3 I did everything, but still my answer is ...
2
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1answer
291 views

Determine the region at xy plane where $(1+3y^3)y´=x^2$ has unique solution at $(x_0,y_0)$

How can i determine the region at xy plane where: $$(1+3y^3)y´=x^2$$ has unique solution at $(x_0,y_0)$?
3
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1answer
54 views

Differentiation calculation

$L(E)$ espace fonction continuous and linear $$\begin{array}{llll} \psi:& L(E)\times E&\longrightarrow& E\\ &(u,x)&\longrightarrow &u(x) \end{array}$$ proved the application ...
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2answers
37 views

Does the Euler-Lagrange equation have a series solution?

In classical mechanics the Euler-Lagrange equation of motion is a linear homogeneous ODE of second order, how come we do not have a series solution like other famous differential equations (Legendre, ...
3
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0answers
29 views

Lowering the power of a linear differential equation.

$$L(x)\equiv x^{(n)}+a_1(t)x^{(n-1)}+...+a_{n-1}x'+a_n(t)=0.$$ The solutions $x_1, x_2,...,x_m (m<n)$ are given. Linearly independent. Let us find $x_{m+1},...,x_n$ It's starts off like this, I ...