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

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3answers
58 views

Lyapunov stability at origin with identically zero test function

At the origin, determine stability of $$x' = y \\ y' = -\tan(x)$$ If we use the test function $V(x,y) = 0.5y^2 + \int_0^x tan(s)ds$, we get $\dot{V}=x'\tan x +y'y = y\tan x -y\tan x = 0$, so the ...
1
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1answer
21 views

Norm bound on exponential matrix with eigenvalue negative real part, proof

If $A$ is $n \times n$ with negative real parts of all eigenvalues, then there exists positive $K,\alpha$ such that $$\|e^{At}\| \leq Ke^{-\alpha t}$$ Furthermore, if an eigenvalue has negative part ...
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2answers
42 views

Euler Cauchy equations, change of variables

To convert an euler cauchy: $x^{2}y''+pxy'+qy=0$ equation into a linear one we perfom the substitution $x = e^z$ from which we get: $$z=\log x$$ $$\frac{\mathrm{d} x}{\mathrm{d} z} = e^z =x $$ ...
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1answer
72 views

How to integrate $\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$?

How to integrate: $$\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$$ I don't really have a clue? Do I need to simplify it first somehow?
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2answers
31 views

Solving $\frac{b}{a-b}e^{at}=\frac{x(t)}{a-x(t)}$ for $x(t)$

I`ve been trying to solve the differential equation $x(t)'=x(t)(a-x(t)), x(0)=b, t\in [0, \infty]$. Using the technique of seperation of variables, I get $\frac{b}{a-b}e^{at}=\frac{x(t)}{a-x(t)}$. Now ...
1
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2answers
33 views

What makes a differential equation, linear or non-linear?

Among these differential equations why one is linear while other is non-linear? What is criteria to find out whether a differential equation is linear or non-linear?
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1answer
27 views

Ordinary differential equation . [closed]

The roots of the auxiliary equation for a homogeneous linear differential equation with real constant coefficients that has $ y= 4 + 2x^2 - e ^{-3x}$ as a particular solution are : 1) $ m= 0 , 0 , ...
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0answers
36 views

Proving this is the unique solution to this simple system of diff equations.

So the set of equations are these $\frac{d \omega_x}{dt}+\Omega \omega_y =0$ $\frac{d \omega_y}{dt} - \Omega \omega_x =0$ You can easily differentiate again, get two second order linear diff ...
1
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0answers
40 views

how to rearrange matrix equation to have unknown in vector form

I am looking for the name/type of following equations: $$\dot{\theta}\dot{J} = \ddot{x} - J\ddot{\theta}$$ here the unknown is $J \in R^{m \times n}$, $x \in R^{m \times 1}$, $\theta \in R^{n \times ...
3
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1answer
28 views

Question about assumptions for Picard-Lindelöf Theorem in Zeidler's functional analysis text

In Zeidler's text on functional analysis pg.24 he wrote... The Picard Lindelöf Theorem: Assume the following: (a) the function $F: S \to \mathbb{R}$ is continuous and the partial derivative ...
5
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3answers
109 views

How to solve this DE?

Consider the ordinary differential equation $$y''=xyy'$$ I'm pretty stumped, so any tips on how to proceed? It seems fairly simple but I'm drawing a blank.
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3answers
64 views

Initial Value Problem $dy/dx = (y+1)^{1/3}$

Consider the differential equation $$\frac{dy}{dx} = (y+1)^{1/3}$$ (a) State the region of the $xy$-plane in which the conditions of the existence and uniqueness theorem are satisfied (using any ...
2
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2answers
30 views

Coupled second-order differential equations

I am trying to solve the following system of coupled ODEs: \begin{align} -x^2 f'' - 3xf' + (1-2a)f - (a+1)x^2g'' + (2-4a)xg' + (4a-2)g &= 0,\\ (a-1)x^2 f'' + (4a+2)xf' + (12-6a)f + 12xg' + ...
1
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3answers
224 views

Solution of Differential equation

Question: Find solution of differential equation $$ 3e^{4x} \frac{dy}{dx} = -16\frac{x}{y^2} $$ which satisfies the initial condition y(0)=1 Solution: I know that I have to bring it in the general ...
3
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1answer
33 views

homogeneous first order differential equation

is there a method to solve $$\dfrac{dy}{dx} = f(x,y)$$, where $f(x,y)$ is a homogeneous function. I found some examples like $f(x,y)=(x+y)^2$ where it can be solved after converting it to Ricatti's ...
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1answer
37 views

graphing the solution of $y'=x^2-3$

I have a Ordinary Differential Equation(ODE) and I got the solution as ​ Now I want to draw graph? How can I do that? I think: ...
5
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3answers
181 views

Given the differential equation, how to solve the y function with x as the independent variable?

$y\frac{dy}{dx} = x(y^4 + 2y^2 + 1)$ $y = 1$ when $x = 4$ I tired to integrate by substitution, but it doesn't seem to work out.
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0answers
18 views

Dimensional reduction of system of ODEs

Given a nonlinear system of eight autonomous differential equations with all variables and parameters living in the positive octant of real numbers: $$dX_1/dt = \ldots\\ dX_2/dt = \ldots \\ \ldots ...
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3answers
48 views

Solving the differential equation $dr=(r\cos\theta +r\sin\theta)d\theta$

$dr=(r\cos\theta +r\sin\theta)d\theta$ In my book this is under separation of variables then i tried to factor out r and divide both sides then integrate both sides but where can i find my $C_1$? I ...
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0answers
62 views

Trapping region for Nonlinear ODE system?

I need to find a trapping region for $u'=-u+vu^2$ $ v'=b-vu^2$ with $b>0$. I don't know what theory to use or in wich book I can find some examples to find optimal trapping regions. Thank you ...
0
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1answer
26 views

Existence of the first weak eigenvalue of the Laplacian in a bounded domain

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $$H:=W_0^{1,2}(\Omega):=\left\{u\in L^2(\Omega):\nabla u\in L^2(\Omega)\right\}$$ be the Sobolev space, where $\nabla u$ denotes the weak ...
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3answers
43 views

Finding the Correct Function that fits the Scenario

i have been trying to find a function that fits the following scenario: $$ f'(c) = 1^0 $$ $$ f''(c) = 2^1 $$ $$ f^{(3)}(c) = 3^2 $$ $$ f^{(4)}(c) = 4^3 $$ and so on, the purpose is to derive a way to ...
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0answers
208 views

How to classify/ solve this PDE?

I am searching how to solve the PDE below but I can not seem to find a decent example online. My major did not focus much in solving PDEs so I feel very deficient. I know how to solve for the steady ...
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0answers
60 views
+50

Jacobi field geodesic and calculus of variations.

How can we show that the second order variation to a geodesic is given by the Jacobi differential equation? In essence, \begin{equation} \frac{D^2}{dt^2}J(t)+R(J(t),\dot \gamma (t))\dot \gamma ...
3
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2answers
34 views

LaSalle invariance, Lyapunov stability

Trying to understand the LaSalle invariance principle. Consider the system $x' = y \\ y' = -y-6x-3x^2$ a) Using the test function $V(x,y) = 0.5y^2+3x^2+x^3$, show that the origin is asymptotically ...
0
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2answers
19 views

Domain of existence for this ODE.

I think this is some pre-calculus concept that I've forgotten. I am supposed to solve this initial value problem and determine how the interval in which the solution exists depends on $a$. $$yy' + x ...
0
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1answer
11 views

How to compute the solution of a differential equation involving Brownian local time

My problem is to compute numerically a function F. F is known to be convex and have kinks. It's also known to satisfy a "second order differential equation". Since the function is not everywhere ...
0
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0answers
16 views

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 ...
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1answer
24 views

Hamiltonian flow local diffeomorphism?

I am currently reading Arnold's proof of the Darboux theorem in his book on classical mechanics and fail to understand some point. The background So he wants to show that any symplectic form is ...
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2answers
62 views

differential equations of second order [closed]

How may I solve this differential equations: $$y''+4y=12x^2-16x\cos(2x)?$$
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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) - ...
1
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1answer
30 views

Stability of a system of differential equations of the form $\dot x = y, \dot y = g(x)$

Let $g: \mathbb{R} \to \mathbb{R}$ be a locally Lipschitz-continuous function with $g(0) = 0$ and $xg(x) < 0$ for all $x \neq 0$. Consider the differential equation $\dot x = y, \dot y = g(x)$. I ...
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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 ...
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 } ...
1
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0answers
30 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), ...
0
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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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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 ...
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 ...
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 ...
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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 ...
0
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4answers
85 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 ...
0
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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
28 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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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 ...
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 $$\frac{dx}{dt}= y $$ ...
1
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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 ...
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 ...