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

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What is the general way to rewrite an ODE with respect to a change in coordinates?

I have an ODE : $y' = f(x, y)$ I change for coordinates $(r, s) = (g(x, y), h(x, y))$. What is the equation like in terms of r and s ? If it can help, in my case, $(r, s) = (y.x^{-k}, ln(x))$. I can ...
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8 views

Differential Equations of Transformed System

At the moment I'm struggling with a problem I found in a script to one of my lectures: Let $\phi \in C^\infty(\mathbb{R}^{2n})$ have the property that the system $p_i=\frac{\partial}{\partial ...
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64 views

A line integral equation popped up when trying to derive Exact ODE integrating factor, can it be solved analytically?

(For convenience, for any functions, only its first instance the x,y dependence will be written out, all subsequent instance the x,y will be suppressed) I have an ODE $$M(x,y)+N(x,y)\frac{dy}{dx}=0$$ ...
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55 views

book suggestion on manifolds

I've to learn differential equations on Manifolds. Can any one please suggest some books/lecture notes for differential equations on Manifolds ?
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45 views

Stability of non-autonomous stochastic differential equation

I'm looking for a good reference or insight to under what conditions can I prove stability (or instability) for the following general n-dimensional non-autonomous stochastic differential equation: ...
2
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30 views

Determining linear independence of three simple functions for a third order ODE. (2.9-7)

This is a very similar post to one previous by me but I felt that not all questions were satisfactorily answered. But I am sincerely grateful to those who tried. I would like a sharp independent eye ...
2
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50 views

suggestion for lyapunov function

Consider differential equation \begin{align}x'&=-t(x+y)\\ y'&=-y+x-y(y^2-6).\end{align} Can some one suggest a lyapunov function for it. I have examined $V(x,y)=x^2+y^2$ , ...
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140 views

Find the critical curves for the following functional

Find the critical curves for the following functional : $$J[y(x),z(x)]=\int_{0}^{1}(y'^2+z'^2-xyz'-yz)dx$$ With the conditions : $$K[y(x),z(x)]=\int_{0}^{1}(y'^2-xy'-z'^2)dx=2$$ $$y(1)=z(1)=1$$ ...
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40 views

Is this a valid equivalence between the classes of Differential Equations?

Consider the general first order Linear Ordinary Differential Equation: $$ \frac{dy}{dx} = A(x,y) = \frac{F(x,y)}{G(x,y)}$$ This equation is characteristic equation of the Partial Differential ...
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43 views

Two linear independent functions have zero Wronski

Suppose $f$ and $g$ are linear independent $C^1$ functions on $[a,b]$ and there Wronski det is zero, i.e. $$fg^{'}-f^{'}g=0$$ Can we say there exist an point $t_0\in [a,b]$ such that: ...
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23 views

Why is the basin of an asymptotically stable solution to a differential equation an open set?

Why is the basin of an asymptotically stable solution to a differential equation an open set? The basin is defined as the set off all points s.t. their limit at $t = \infty$ is equal to some solution ...
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66 views

Pursuit Curve, Parametric Equation

So its a classic problem: Object A starts at the origin (0,0) and moves straight up the y axis with a speed v. Object B starts at point (1,0), always moves towards object A and has a speed of 2v. ...
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83 views

a system with differential equations

I have a system which is described by the following differential equation. I want a closed form formulas to calculate $v_1(t)$ and $v_2(t)$ with the given parameters. In the following equations $p, k, ...
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61 views

Recursive differential equations

Suppose we took the odd solution to $y''+y=0$ which is $\sin(x)$. If we put this in place of $y$ in the differential equation we get the equation: $$y''+\sin(y)=0$$ the odd solution to which is an ...
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146 views

best book for practicing unsolved problems in differential equation and linear algebra

I started reading differential equation and linear algebra. Can anyone provide the link/book name where I may get many questions to practice. Generally, in the end of book only few problems are there. ...
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61 views

Equilibrium points and linear stability

Consider the nondimensional amplitude equation for $A = A(t)$ where $t$ is time given by (1): $$ \frac{dA}{dt} = \sigma A - a_1 A^3 - a_3 A^5 = f(A) \text{ with } \sigma \in \mathbb{R}, a_1 < 0, ...
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51 views

Why does this nonlinear ODE solution not work?

I am relatively new to Python and trying to use it to solve a second order nonlinear differential equation, specifically the Poisson-Boltzmann equation in an electrolyte. $$\phi''(r) + \frac2 ...
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248 views

Identifying Hamiltonian Systems with Phase Portrait

the following is a homework question (that isn't going to be graded) and I'm not sure how to do it. I know that the solution trajectories cannot cross the H(x,y)=constant curves, but I'm not sure ...
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39 views

BVP eigenvalue problem

I am working on the following problem and I am completely stuck: Show that the eigenvalue problem $$ -u''+4\pi^{2}\int_{0}^{1} u(x)\,dx=\lambda\,u $$ with $u(0)=u(1)$ and $u'(0)=u'(1)$ has ...
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33 views

Stability of an equilibrium solution with 0 denominator

I'm testing the equilibria of a differential equation and found that one has a 0 denominator. Example: $$\frac{dx}{dt}=2x^{(1/2)}-5$$ Which, when you try and evaluate the derivative at 0, you end up ...
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132 views

Stuck trying to solve wave equation in $n$-dimensions.

Solving the wave equation $u_{tt} = c^{2} \Delta{u}$ subject to $u(0,x) = f(x)$ and $u_{t}(0,x) = g(x)$ gives us d'Alembert's formula. I'm looking to solve the wave equation, subject to these same ...
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57 views

solving this differential equation for $y$, Is it even possible?

Lets say I have the following: \begin{gather} \frac{(y')^3 + 3 y' y'' + y'''}{(y')^2 + y''} = \sqrt{1+(y')^2}\\ \frac{((y')^3+3y' y'' + y''')^2}{((y')^2 + y'')^2} = 1+(y')^2\\ \frac{(y')^6 + 6 (y')^4 ...
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36 views

Initial conditions to solve an ODE?

Given is the following inhomogenous linear ODE (4th order): $$q_0\cdot\sigma + q_1\cdot \dot\sigma + q_2\cdot \ddot\sigma + q_3\cdot \dddot\sigma + q_4\cdot\ddddot\sigma = p_0\cdot\epsilon + ...
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45 views

Solving $2y'''(t)+3t\ y(t)=0$.

For a certain problem, I am trying to solve the ODE $$2y'''(t)+3t\ y(t)=0$$ I am pretty clueless what to do here, any hint would be appreciated. Thank you very much.
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30 views

Differential Equation. VERY small problem

I want to ask a question later, after I show you this TESTING: x^2 = 1 Differentiate both sides 2x = 0 TESTING: x = 1 Differentiate both sides dx/dx = 0 1 = 0 So when can I differentiate both ...
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25 views

How to find Laplace transform of a differential equation?

$y′′ + 3y′ + 2y = f$ , $y(0) = 0$ , $y′(0) = 1$ where $f$ is given by $f(t) = \sum_{n=1}^\infty \delta(t−n)$; find a 1-periodic function $y_*$ with $\lim_{t\rightarrow \infty} |y(t)−y_*(t)| = 0$. I ...
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71 views

How to solve system of stochastic differential equations?

I have the following two SDEs $$dN_1=(2a-1)pN_1dt+\alpha_1 N_1dW_1$$ $$dN_2=(2pN_1-\mu N_2)dt+\alpha_2 N_2dW_2$$ $W$ is the standard Brownian motion/Weiner process. This isn't homework, I'm just ...
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113 views

Lyapunov function for a damped pendulum

The question is about damped pendulum. There are two statements I don't understand or I'm not sure if my justification for them is correct. Could you say if I'm right? The example is from a German ...
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80 views

Second order differential equation, power series method

Solve the differential equation $$(x+2)y''-xy'+(1-x^2)y=0 ; \quad X_0=1$$ using the power series method about the point $x_0=1$. I get to this step after deriving the derivatives of the ...
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How is it possible to continue solutions for a differential equation along t?

Given the equation $y' = e^{\sin y+t} + t\cos(y)$. I rewrote it as $$ y(t) = y(0) + \int_{0}^{t}ye^{\sin y+t}+t\cos y $$ I'm asked to prove that every solution can be continued for every t. I know ...
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Study of a system of differential equations

I'm asked to study everything that is possible to know about the sytem$$\begin{cases}x'=x^2-y^2\\y'=2xy\\z'=-z\end{cases}$$ My questions here is, how much can be know about it?, how do I know I ...
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43 views

Find an integrating factor such that $y'=\frac{1-x+y}{x-y}$ is exact

Yet another question of this sort, and hopefully the last. In the previous question I posted, we were lucky enough and the integrating factor was a function of only one variable, the ansatz $\mu_y=0$ ...
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56 views

Annoying differential equation involving composition

Upon trying to crack into a problem, I managed to end up with the following differential equation. $$ y = xy' - y'\circ y', \qquad\text{or}\qquad y(x) = x\cdot y'(x) - y'(y'(x)) $$ I haven't a clue ...
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66 views

Solution of differential equation $x'=Ax$ where $A=PJP^{-1}$

Let $A$ be a $n\times n$ matrix. Suppose $A=PJP^{-1}$ being $J$ the Jordan form of $A$. Prove that if $x(t)=(x_1(t),\dots,x_n(t))$ is a solution for $x'=Ax$, then ...
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Find the 3rd order DE whose general solution is $ y= C_1e^{2x} + C_2\cos x + C_3 x\sin x $

My attempt $$ \begin{matrix} y &=& C_1e^{2x} &+& C_2\cos x &+& C_3x\sin x\\ y' &=& 2C_1e^{2x} &-& C_2\sin x &+ &C_3(\sin x &+& x\cos x)\\ ...
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35 views

A basic ODE question

Let $G \subset \Bbb R^d$ be open and let $ V: G \to [0, \infty)$ be such that $\dot{V} = \nabla V.h : G \to \Bbb R$is non-positive. We assume that $H=\{x: V(x) =0\}$ is equal to the set $\{x: ...
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157 views

Famous parametric curves that are solutions to differential equations

I know that the cycloid satisfies the differential equation $ \left( \frac{dy}{dx} \right)^2 - \frac{2r}{y} + 1 = 0. $ Are there other famous plane curves that are also solutions to a differential ...
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72 views

Yet Another Differential Equations Problem

I come from a non mathematical background, so solving differential equations is something that I have to acquire on the go. I hope the following makes sense. I want to chose a nonnegative ...
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172 views

Fundamental Matrix

Determine $\phi(x,0)$ for $A(x)=\begin{pmatrix} -1 & \cos(x) \\ 0 & -1\end{pmatrix}$, where $\phi(x,0)t_{0}$ is a solution of $\frac{d}{dx}t(x)=A(x)t(x)$. I am not entirely sure as to ...
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66 views

Fundamental Matrices for Linear ODE

Why is the following statement true?: For a matrix ODE: $\mathbf{x'=Ax}$ with special fundamental matrix, $\Phi (t)$ or $e^{\mathbf{A}}$, where $\Phi(t_0) = I$, and fundamental matrix containing the ...
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96 views

An Ordinary Differential Equation with time varying coefficients

Let $A$ and $B$ be complex numbers, let $\beta_1$ be real and $\beta_2=2$. Consider a following Ordinary Differential Equation: \begin{equation} \frac{ d^2 r_t}{ d t^2} + \left(\frac{A}{t^{\beta_1}} + ...
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82 views

Finding Second Solution for Hermite Differential Equation through reduction of order

One can use the ordinary power series solution to find one solution of the Hermite Differential Equation $$ y''(x) - 2 x y'(x) + \lambda x = 0$$ Can one use the reduction of order technique to find ...
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25 views

Linear Differential Equation Initial Value Problem

Below I have the question and my steps to my current answer. Would this be correct? Thank you $$\frac{dy}{dt}+y=11, y(0)=5$$ $$y'+y=11$$ $$p(t)=1 \rightarrow e^{\int p(t)dt}=e^t$$ ...
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47 views

Find behavior near fixed point beyond linear expansion

this is my first question on math.stackexchange, I hope to have phrased it correctly! I have a differential equation $\text{$\frac{\text{d}x}{\text{d}t} = \alpha t^{-3}\frac{f'(x)}{f(x)}$ with ...
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26 views

Is there such a thing as a separable second order ODE?

Say I have a function $y''(x) = g(x)y(x)$. Could I separate this by dividing by $y$ and then integrating? i.e. $dy^2/y = g dx^2$ or maybe $y'/y\ dy = g (dx/dx) dx = g dx$? If so, how does one ...
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51 views

Dixmier Conjecture

In algebra the Dixmier conjecture, asked by Jacques Dixmier in 1968, is the conjecture that any endomorphism of a Weyl algebra is an automorphism. Which are some consequences of Dixmier conjecture ...
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119 views

Mean Value Property for Harmonic Functions (clarifying Axler's proof)

I'm going through Axler's proof of the mean value property, and I'm a little puzzled. Since this proof is in the first few pages of his book, I thought I'd ask for clarification before going further ...
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65 views

List of ODE's that can be solved by Fourier transform

I am teaching introductory level Fourier analysis and I want to give my students some basic and some not so basic examples of how to solve ordinary differential equations with the method of Fourier ...
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42 views

Nondimensionization of a simple system.

A damped spring mass system is modelled below: $$m\frac{d^2y}{dt^2}=F_s+F_d\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space t>0$$ ...
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31 views

Find the initial movement of a particle

A particle with mass $m$ is moving along a curve and the force exerted on it always points towards the origin, and it´s magnitude is proportional to the distance between the particle and the origin, ...