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

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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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81 views

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\} \\ ...
3
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73 views

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 ...
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40 views

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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52 views

Arnold ODE Problem

Problem 1 of Section 1.2.4 of Arnold's ODE book asks, "Can the integral curves of a smooth (continuously differentiable) equation $\frac{dx}{dt} = v(x)$ approach each other faster than exponentially ...
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52 views

Sources on Jacobi Elliptic Functions

I'm interested in learning more about the Jacobi Elliptic Functions and the associated theta functions. For instance, what was the initial motivation for defining them? What are some applications? Is ...
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52 views

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 + ...
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47 views

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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37 views

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 ...
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69 views

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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72 views

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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97 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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34 views

What allows us to break up $dy/dx$ in solving a separable differential equation?

Suppose you had a separable first order differential equation that can be written as $$ \frac{dy}{dx}=f(x,y)=g(x)h(y) $$ Rigourously, what allows us to rearrange this as $$ \frac{1}{h(y)}dy=g(x)dx? ...
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31 views

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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50 views

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 ...
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46 views

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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49 views

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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23 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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64 views

Existence theorem on weak solutions of ordinary differential equations

Consider an ordinary vector-valued differential equation of the form $$ \begin{align*} \dot y(t) &= f(t,y(t)), \\ y(0) &= y_0 \in \mathbb{R}^n. \end{align*} $$ It is well known that if $f$ is ...
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42 views

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 ...
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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} = ...
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81 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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100 views

Non-Linear Ordinary Differential Equation in Fluid Dynamics

So while trying to model the physics of a rocket shot from the ground through the atmosphere, I came up with a second-order Non-Linear ODE of the form: $$ \ddot y + \dot y^2 e^y = f(t) $$ This is ...
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105 views

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 ...
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65 views

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 ...
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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 ...
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62 views

Door mechanism differential equation

I have been wondering about a door mechanism I have seen. It has a wire attached to the upper corner of the door and from there to the corresponding corner in the door frame, where a weight hangs from ...
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64 views

How do I solve this differential equation?

$y^{(7)}+4y^{(6)}+8y^{(5)}+9y^{(4)}+8y^{(3)}+8y^{(2)}+8y^{(1)}+4y=e^{-x} (5sinx-cosx) $ The characteristic equation $ \lambda ^7 +4\lambda ^6+8\lambda ^5+ 9\lambda ^4 +8\lambda ^3+8\lambda ...
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Lipschitz function in ODE

Let $f:\mathbb{R} \times \mathbb{R^n} \rightarrow \mathbb{R^n}$ lipschitz function. Show given that $(t_0,x_0)\in \mathbb{R} \times \mathbb{R^n} $ there exists a unique solution: $$x' = f(t,x)$$ ...
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126 views

Fundamental Solution of a Nonlinear ODE (using Riccati Transformation & Wronskian)

I am given the differential equation: \begin{equation*} y^{\prime}(t) = y(t)^{2} + 2\sin(t)\cos(t) - \sin^{4}(t) \end{equation*} and one solution $y_{1}(t) = \sin^{2}(t)$. I wish to find a second ...
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100 views

Cauchy-Euler Equation of order $n$

What I wish to prove is that for a Cauchy-Euler equation of order $n$, the substitution $x=e^{t}$ transforms it into a linear differential equation with constant coefficients. To put it as a theorem: ...
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84 views

Differential Equation has a unique solution periodic

Let $A(t)$ continuous and periodic of period $S$ in $\mathbb{R}$. Suppose $x' = Ax$ has $\varphi \equiv 0$ as the only periodic solution of period $S$. Show that there exists $\delta> 0$ such that ...
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Invertible e converges series.

If T is a linear transformation on $R^n$ with $||T - I||<1$, prove that $T$ is invertible and that the series $\sum_{k=0}^\infty(I-T)^k$ converges absolutely to $T^{-1}.$ (Use the geometric series) ...
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108 views

WKB and asymptotic behavior of second order differential equation

I want to study the large $x$ solution to a Riccati equation. After listening to the lectures on Mathematical Physics by Carl Bender, I have fallen in love with asymptotic analysis. But, by no means ...
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167 views

Are any solutions lost when solving non-exact differential equations?

I have just started studying differential equations, one of the problems I found while I was practicing is "Consider the equation $$ (y^2 + 2xy)dx - x^2 dy=0 $$ (a) Show that this equation is not ...
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108 views

Solve ODE by Fourier transform, and versus by Laplace transform?

Regarding solving ODE by Fourier transform, I read a nice reply by O.L.. After applying Fourier transform to an ODE to obtain an algebraic equation, the reply showed that some terms involving the ...
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53 views

Show that this initial-value problem has a unique solution

I am trying to show that the following initial-value problem $$\frac{dx}{dt} = - x + tx^{1/2}; \quad x(2) = 2$$ has a unique solution on $I = [2,3]$. By letting $f(t,x) = - x + tx^{1/2}$ and $(t_0 ...
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How to integrate/differentiate parameters in differential equations

I have an ODE, which I would be fine about solving, were it not for the parameter: $$(\omega^2+x^2)\frac{dy}{dx}=y$$ I'm given that $\omega>0$ is a parameter. Separating the variables gives: ...
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46 views

Would the transformation of a differential equation obey the same algebra?

I've found that the algebra of this differential equation $$\frac{d^2y}{dz^2}-(3z^2+\gamma)\frac{dy}{dz}+(cz+\alpha)y=0$$ is in $sl(2)$ because it is possible to use the generators of the $sl(2)$ ...
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Measurability of points regular

I'm reviewing the proof of the theorem of oseledet the book Mañe: Let $M$ a compact metric space and $f:M \rightarrow M$ a homeomorphism, $\pi: F \rightarrow M$ a finite-dimensional continuos vector ...
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112 views

Does the implicit function theorem imply Peano existence theorem

In The implicit function theorem written by Krantz & Parks, it's said that the implicit function theorem implies the following existence theorem of ODE: Theorem 4.1.1 If $F(t,x)$, ...
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49 views

Qualitative dependence of solution to second-order matrix differential equation on eigenvalues

Suppose we have a matrix differential equation in $\vec{x}(t)=\left(\begin{smallmatrix}x_{1}(t) \\ \vdots \\ x_{n}(t)\end{smallmatrix}\right)$, such that: ...
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ODE with delta function

Consider the following ODE $$y''+a\delta (x)y+\lambda y=0$$ subject to the initial conditions $$y(\pm\pi )=0$$ (1) Show that there is a set of eigenvalues $$\tan (\pi \sqrt{\lambda ...
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Can't match boundary conditions on a perturbation series solution to a non-linear ODE?

I'm trying to generate a naive perturbation series solution (with all associated secular terms included) to the Rayleigh equation: \begin{equation} \frac{d^2y}{dt^2} + y = \epsilon ...
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55 views

Merton's Problem Stochastic Differential Equation

Solve the following numerical case of Merton's optimal portfolio selection problem: find an optimal policy function $(s, y) \mapsto u(s, y)$ such that for the Ito diusion determined by $dX_t =X_t(u(t, ...
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76 views

solve nonlinear second order ODE

I obtained Nonlinear second order differential equation as $y\cdot y''+y'^2-m\cdot y^{-a}y'^2+k=0$, Where $y'= \dfrac{dy}{dx}$, $y''=\dfrac{d^2y}{dx^2}$. I could not obtain the solution so please ...
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144 views

Lyapunov Function and $\omega$-limit sets

I will ask you about a particular equation but what I would really enjoy is an (if possible, comprehensive) answer to the following question : How can we, using a Lyapunov function, study the ...