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

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535 views

Finding the modified Green function for the Helmholtz equation

I've been wrestling with this question for quite some time now, and the result was like 20 leaves of paper packed with scribbling...anyway, here's the question: I need to find the solution to the ...
2
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15 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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31 views

Master equation of chemical reaction

I have about the construction of master equation for chemical reaction i.e. I have to construct differential equations for the probability mass function for the number of particles A, B and C. When ...
2
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0answers
38 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 ...
2
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30 views

Finding the flow of a pushforward of vector field (small edit needed as well)

Let $\mathbb{X}$ be the vector field on $\mathbb{R}^2$ given by $$ \mathbb{X}(x,y) = (y,x). $$ Compute the flow $\Phi_t$ of $\mathbb{X}$. Let $F:\mathbb{R}^2\to \mathbb{R}^2$ be the ...
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13 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 ...
2
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31 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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19 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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37 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 ...
2
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54 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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43 views

Riccati differential equation

The Riccati differential equation, $y'=x+y^2$ is special equation. I know that how can I solve it, but my problem is that I don't have initial conditions, and I firstly need a particular solution. How ...
2
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27 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 + ...
2
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39 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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28 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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43 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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43 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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15 views

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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57 views

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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34 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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42 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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45 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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28 views

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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33 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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26 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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33 views

What are the main similarities between difference equations and differential equations, specifically in methods for solving them?

I know that that difference equations can be used to represent discrete dynamical systems and differential equations can be used to represent continuous dynamical systems. Therefore both are ...
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50 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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32 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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29 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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22 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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24 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 ...
2
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0answers
86 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 ...
2
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0answers
44 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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21 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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19 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, ...
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41 views

Period of a pendulum

Consider the pendulum problem $\frac{d^2x}{dt^2}+\sin(x)=0$ $\frac{dx}{dt}(0)=v_0=0$ $x(0)=x_0$ Show that the period ...
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16 views

2nd order linear differential equation with non-constant coefficients

Considering the equation $2y''+(x+1)y'+3y$ where $X_0=2$. Find the general term in each solution. That is, the general term for Y1,Y2 where $y=A_0(Y_1)+A_1(Y_2)$ I've solved this as ...
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0answers
41 views

Validity of approximating a difference equation with a differential equation

Consider the following difference-differential equation defined for positive integer indices $k$ and $t$: $$ A_k(t+1)-A_k(t)=\beta \frac{ A_{k-1}(t)- A_k(t)}{\alpha+2 t} + \delta_{k \beta} . \tag{1} ...
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24 views

Fluid Flow: lubrication, integration, ODE

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
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0answers
15 views

Derivation of higher order bessel function in terms of lower order functions

I am really stuck trying to prove this.. ((x^-p)Jp(x))’ = -(x^-p)Jp+1(x) ---(1) Can someone please help how to actually prove this step by step, because whichever notes i see, they prove ...
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0answers
25 views

Solve the following ODE

Solve the following ODE $$(y-x)\left(1+x^2 \right)^{\frac{1}{2}}\dfrac{\mathrm{d}y}{\mathrm{d}x}=n\left(1+y^2 \right)^{\frac{3}{2}}$$ I have tried substituting $y=\tan \theta$ and $x=\tan \phi$ ...
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69 views

Prove that solutions to linear system form a vector space of dimension $\geq 2$

I accept & appreciate any form of help with the following problem: $B_{nxn}$ "periodic matrix" with period $T$ such that $B(t+T) = B(t)$ for all $t\in \mathbb{R}$. Assume that the system $x' = ...
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0answers
46 views

Show that $\displaystyle\sum_{i=0}^{N-1}|\epsilon_i|\to0, N\to\infty$

Let $I_o=[t_0,t_0+T]\subset\mathbb R, T>0$, If $f\in C^0(I_0\times\mathbb R,\mathbb R)$ and satisfies the Lipschitz condition: $\forall t\in I_0, \forall y,y^{*}\in\mathbb ...
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0answers
30 views

Geodesics on a perturbed submanifold of $\mathbb{R}^m$

Let us consider $M$, a Riemannian manifold of dimension $n$, isometrically embedded in $R^m$. Let us consider a geodesic $\gamma$ on $M$. Now, let us "perturb" (in other words, change slightly the ...
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26 views

How to define a Holder seminorm of a section

I'm reading "Variational Problems in Geometry",Seiki Nishikawa, in the figure below. Let $(M,g)$ be a compact $m$ dimensional Riemannian manifold with no boundary. $T>0, 0<\alpha<1, ...
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34 views

How can I prove these two fields are locally topologically conjugated?

The problem is to prove that the fields $x'=x$ and $x'=x^3$ are locally topologically conjugated in the origin. I found that the corresponding flux for the first equation is $\phi(x_0,t)=x_0e^t$.The ...
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60 views

Math software for plotting phase portraits

I'm looking for math software which is possible to plot phase portraits for ODE and systems of differential equations. Is there a software which can create not only simple 2D phase portrait plots but ...
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0answers
38 views

Nontrivial solutions for a system of equations

Consider $t:[0,1]^2\to R$ that is differentiable a.e. and satisfies conditions (i)-(ii): (i) $$ \int_0^1 \frac{\partial t}{\partial t_1}(x,y)f(y\mid x)\,dy=0, \quad \forall x\in[0,1] \\ \int_0^1 ...
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39 views

How to classify the equation $\frac{dy}{dx} + x^{2}y = xe^x$

I have the following "homework" problem: Classify each of the following differential equations as ordinary or partial differential equations; state the order of the equation; and determine ...
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23 views

difference between runge kutta methods of same order

I recently read about runge kutta methods for solving differential equations. So far I understood the idea but up to know nobody could answer me following question: If we consider the explicit rk ...
2
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0answers
50 views

Understanding the eigenvalue problem: $x^2y''+xy'+\lambda y = 0$

I would just like to clarify a fw things I am not really understanding about Sturm-Liouville forms and eigenvalue problems: I have the practice question: $x^2y''+xy'+\lambda y = 0$ with boundary ...