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

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Center Manifold Exercise (small solution for small changes of the parameter)

Hi I'm stuck with this problem at first I didn't know how to begin so I copy an argument from [Carr, Application of Centre Manifold Theory]. But I don't know how can I find the coefficient from a, b ...
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212 views

Differential Equations: Jordan Form of a Matrix

I am using Lawrence Perko's book Differential Equations and Dynamical Systems, for my Differential Equations course. At the moment we are going over Jordan Forms of a linear system $x^{'}(t) = Ax$, ...
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95 views

Proofs from Ch. 1 of Arnold's ODEs

I've started reading Vladimir Arnold's Ordinary Differential Equations on my own. I like it so far, the only problem is that all of the exercises (as yet) are of the type "prove $X$" and without an ...
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51 views

Squares and constants in the dynamical system

I have $$ \begin{eqnarray} x'&=& x^2 - y^2 -1 \\ y'&=& 2y \end{eqnarray} $$ How can I solve such a system? I have tried the substitution $X= x^2 - 1$ but I still get constants in ...
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Coupled partial differential equation, with boundaries specification

Please, help me to find a books or samples to learn how to solve such coupled equations $$\begin{eqnarray} \frac{\partial T_1(x,t)}{\partial t}&=& \alpha_1 \frac{\partial^2 T_1(x,t)}{ \...
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80 views

Green's Functions: Solvable non homogeneous Sturm-Liouville with non homogeneous boundary conditions

I was just presented with this problem in my PDE Methods course which involves a non homogeneous Sturm-Liouville problem, which states as follows: Find the conditions under which the following SL ...
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50 views

Stability of origin of dynamical system

Usually you can note some nice structure in the problem which enables construction of a nice Lyapunov function. But this one is just a monster. Maybe there is a trick I've missed? Investigate the ...
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56 views

Reducing a higher order nonlinear ODE to a system of first order ODEs

The ODE that I am trying to reduce is: $$ y''' + 4\,y'' + y' + 6\,y - 2y^{2} = 0 $$ I start by letting $$ y = y_1 $$ $$ y' = y_2 $$ $$ y'' = y_3 $$ $$ y''' = y_4 = 2y_1^2 - 4y_3 - y_2 - 6y_1 $$ ...
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Difficulty understanding Floquet multipliers wrt Mathieu equation

We have the system $$\begin{pmatrix}y\\z\end{pmatrix}' = \begin{pmatrix}0 & 1 \\ a-2\epsilon \cos t & 0 \end{pmatrix}\begin{pmatrix}y\\z\end{pmatrix}$$ and from Abel's formula we have that $\...
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Using a Fourier Series to Solve Differential Equation

The problem states to use the fourier series of the function f(t) defined as follows: $f(t)= t+1 , -1<t<0 $ $f(t)=1-t , 0<t<1$ to solve the differential equation: x''+4x=f(t), x(0)=1, ...
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Which functions solve autonomous ODEs?

Fix an open set $U \subseteq \mathbb{R}$. What can be said about the set $a_n(U)$ of functions $f \in C^n(U, \mathbb{R})$ for which there exists a (sufficiently nice) nonzero function $g : \mathbb{R} ...
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Differential Equations which involve Infinite Series

The problem statement is as follows: Find the general solution for the following equation for $x(t)$. $$x''+ 9x = 2 + \sum_{n=1}^\infty \cos(nt)/n^3$$ I can't find anything about this in my ...
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Find extremum of functional

I want to find the extremum of $$J(y)= \int_1^2 \frac{\sqrt{1+y'^2}}{x}dx, \ y(1)=0, \ \ y(2)=1$$ I thought to use the following theorem: If $y$ is a local extremum for the functional $J(y)= \int_a^...
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Uniqueness of Rectifying Coordinates: Question for Arnold's ODE Book

In section 7 of his book Ordinary Differential Equations, VI Arnold explains the `rectification theorem', that, given an ordinary differential equation $$\dot{\mathbb{x}} = \mathbb{v(x)}$$ where $\...
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Solve $x(x-1)y''+6x^2y'+3y=0$ using Frobenius's Method

Solve $x(x-1)y''+6x^2y'+3y=0$ using Frobenius's Method I can't solve this ODE. How can I get first two term? and indicial equation is also very confusing. I can solve two term recurrence relation ...
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51 views

Asymptotic Behavior of Differential Equation

physicist here. I'm studying some problems that involve the use of differential equations. The professor of the course has indicated that usually variable changes used to simplify the equations come ...
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What type of equation is this?

Is this equation an ODE or PDE $$ \frac{d^3u}{dx^3}−αxu=0, x∈R $$ The only thing given is $\int_R u(x) =\pi $ and $α>0$ is some constant. I have to find the solution using fourier ...
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85 views

If $y'=\frac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) $

If $y'=\dfrac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) = ?$ By integrating I am getting $$y = \ln (x+1)+C$$ I am stuck somewhat as it looks tricky from here. Any help ? Thanks!
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396 views

Solving a differential equation with a square root

I am trying to solve the differential equation $ A(x)\frac{d^{2}f(x)}{dx^{2}}+B(x)\frac{df(x)}{dx}=\frac{1}{3}\frac{1}{\sqrt{f(x)}}, $ where $ A(x)=\frac{x}{x+1} $ and $ B(x)=\frac{2x+1}{(x+1)^{2}} ...
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203 views

Wave equation for a string nonuniform (PDE)

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman , but I have been impossible these paragraphs. The displacement $u$ of a nonuniform string ...
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100 views

Using Green's Function to Find Particular Solution

We have the non-homogeneous differential equation $x^3y'''-3x^2y''+6xy'-6y=4x^2$ with conditions $y(1)=1, y'(1)=1, y''(1)=0$, and I have been tasked with finding its particular solution using Green's ...
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Existence and uniqueness of strong solution of stochastic differential equation.

I am currently going through the proof of the existence of a solution of the SDE \begin{align} dX_t = bdt + \sigma dB_t \end{align} where $B_t$ is a Brownian motion with respect to a filtration $\{\...
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297 views

Applied differential equation regarding water clocks

We have a water clock, the shape defined by $r=f(h)$, and the time marks on this water clock are equally spaced. We have to find f(h), and graph $h$ as a function of $r$, assuming the hole through ...
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Writing ODE system with a complex variable

I'm looking at the system of ODEs: $$\begin{cases}\dot{x} = -y + kx + xy^2\\ \dot{y} = x + ky - x^2\end{cases}$$ I'm trying to introduce a complex variable $z = x+iy$ to write this as a single first ...
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Solution for an ODE given only at discrete points

The problem I have: For each $n \in \mathbb N$ I have $$\begin{align} x_0^n & \in \mathbb R \\ h_n & \in \mathbb R \\ x_k^n & = x_0^n + k \cdot h_n \text{ for } k \in \{0,1,\ldots n\} \\ ...
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Voltera equation

Consider the Voltera integral equation: $$ψ(x)=e^{-x}\cos(x)-\int_{0}^{x}e^{-(x-t)}\cos(x)ψ(t)dt$$ How can I solve this equation by converting it to a differential equation? The solution is $$ψ(x)=\...
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A question about solving the nonlinear differential equation $\dot{x} = x(1-x)$

I am aware of the standard solution that makes use of partial fractions. However, I made the following manipulations, in order to be more rigorous with splitting up the differentials before ...
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Solution of inhomogenous ODE (4th order)

Hello stackexchangers, I have an inhomogenous ODE in 4th order. This ODE is the constitutive law to describe a material by using the "Wiechert model" (p. 15) which is given by $p_0\sigma + p_1\frac{\...
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Kinematics of gravity in a non uniform field

I am a first year physics student. I am trying to figure out how to compute position in terms of time for an object falling through non uniform gravity towards the earth, and by extension towards any ...
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the exact graph of the general solution for $x'=\begin{bmatrix} 1 & 1\\ 4& 1 \end{bmatrix}x$

i need someone to give me exact graph of the general solution for $$x'=\begin{bmatrix} 1 & 1\\ 4& 1 \end{bmatrix}x$$ i solved it manually , the general solution is like this $$x(t)=c_1\...
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Existence of Periodic Solution

I'm working with the system of equations below that represents a Pendulum with constant forcing. \begin{align*} \theta'&=v\\ v'&=-bv-\sin(\theta)+k \end{align*} Where $\theta$ gives the ...
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How to adapt the discrete-time to continuous, $(A) \Rightarrow (B)$?

in class was proved oseledets theorem for discrete time, following guidelines Ricardo Mañe book. Theorem discrete Oseledets (A) : Let $ M ^ n $ be a Riemannian manifold, $ f: M \rightarrow M $ be ...
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237 views

How can one derive Stokes lines of the Stokes phenomenon of asymptotics from a differential equation?

Is there a standard technique to calculate Stokes lines and anti-Stokes lines of the Stokes phenomenon of asymptotics for a function defined as the general solution to a differential equation without ...
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Second order DE question

I am looking for tips for this equation: $ 4xy''+y'+xy'+\frac{3}{2}y=0 $. I am solving with the substitution y=a(x)b(x), but it is getting messy..
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Finding the general solution to system of linear equations: $y' = 2y,y''=4y-y'$

Question: I want to find the general solution to the following system: $\begin{pmatrix} \dot{y}_1 \\ \dot{y}_2 \end{pmatrix} = \begin{pmatrix} 2&0\\4&-1 \end{pmatrix}\begin{pmatrix}y_1\\y_2\...
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General Solution of DE?

I've got the following ODE, and I'm just having trouble coming up with the form of the general solution. I'm really trying to find the particular solution, but in order to do that, I need to know the ...
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287 views

Combining two differential equations

I have two differential equations that are connected by an equation, $L_1\frac{d^2I_1}{dt^2} + \frac{1}{C_1}I_1=\frac{dV}{dt}$ $L_2\frac{d^2I_2}{dt^2} + \frac{1}{C_2}I_2=\frac{dV}{dt}$ $I_1+I_2=I$ ...
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Find a series solution to $(x^2-2)y''+6xy'+4y=0$.

Find a series solution to $(x^2-2)y''+6xy'+4y=0$. A. Find the recurrence relation to $a_n$: My answer is $a_{n+2}=a_n\cdot \frac{n+4}{2(n+2)}$ which is correct. B. Using A, write two independent ...
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27 views

Differential Equations: Say whether the equation has bounded solutions at $x = 0$

Its been forever since I've done Diff EQ and I can't remember how to go about solving this problem: Say whether the equation has bounded solutions at $x = 0$ and whether all solutions are bounded: $$ ...
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Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
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Solving a system of first order differential equations

So, I have (another) problem with differential equations (from an optimal control problem). I am trying to solve the following system of DEs (is this even a system?): $$ \lambda'(t) = r \lambda(t) + ...
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Let $\eta (x)=\int_0^\infty e^{at}\xi(\phi_t(x)) dt$ then $\eta$ is a $C^1$ function

Consider the following problem. Suppose that $a>0, r >0$ and $\xi:\mathbb R \to [o,\infty)$ is a $C^2$ which vanishes in the complement of the interval $(-r,r)$. Also suppose that $\xi(0)=\xi'(0)...
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What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = kf_{...
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85 views

Ordinary differential equation­

$$\dfrac{dy}{dx}-\dfrac{\tan y}{1+x}=(1+x)e^x\sin y$$ I tried $\sin y=t$ but failed. It seems to immune to methods I know of or I am just unable to make the right substitution... Wolfram alpha ...
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Solving 1D telegrapher's equation by reduction to two-dimensional wave equation

The solution $w : \mathbb R \times \mathbb R_{+} \to \mathbb R$ of the Cauchy problem for the telegrapher's equation $$ w_{tt} - c^2 w_{xx} + c^2 \lambda^2 w = 0 $$ with $$ w(x,0) = 0, \qquad w_t(x,...
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Solution to Schrödinger equation $ \partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t)V(x)f(x,t).$

I want to solve $$ i\partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t-t_0)V(x)f(x,t),$$ for any $V \in C^{\infty}[-1,1]$ and $f: [-1,1] \times \mathbb{R_{\ge 0}} \rightarrow \mathbb{C}$. I would ...
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Finding the exact solution of a differential equation

Let $y=f(x)$. Is it possible to find an exact solution of the following differential equation?: \begin{equation} \ddot y+2\dot y-5xy=e^{-2x}\nonumber \end{equation} Many thanks in advance, -- Cesar
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Let $Z=Z(x,y)$ be a solution of $\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}$ = 1

Let $Z=Z(x,y)$ be a solution of $$\frac{\partial z}{∂x}\frac{\partial z}{\partial y} = 1$$ passing through $(0,0,0)$. Then $Z(0,1)$ is 0 1 2 4 By Charpit Method I get the solution ...
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204 views

Analytical solutions of Thomas Fermi equation

The Thomas Fermi model of atoms and nuclei is used in many applications of atomic and nuclear physics. The ODE related to this model is: $$\frac{d^2}{dx^2}\phi(x)=x^{-\frac{1}{2}}\phi(x)^{3/2}$$ with ...
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45 views

Green's function the way George Green defined it

This is a curious question about the way George Green could have defined his Green's function. All the definitions I see have only Dirac-delta $\delta(x-x')$ function as their source on the RHS. But ...