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

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Picard's uniqueness theorem

The differential equation $$\frac{dy}{dt} =ay; y(0)=0$$ satisfies the condition of Picard's uniqueness theorem. But, I find that it can have multiple solution. How lipschitz condition helping to get a ...
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38 views

Calculation of the coeffiecients

I am solving an initial-boundary value problem and I got to a point that I have to calculate the coefficients knowing that: $$(1-b)x-a=\sum_{n=1}^{\infty}{A_n \sin{(\frac{(2n+1) \pi x}{2L})}}$$ One ...
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25 views

Relation between limits of related functions?

I have a function (a differential equation) $$\frac{dP}{dG}$$ which I can use to optimise a system. I.e., set $dP/dG=0$ and solve numerically for $G$. $P$ is also a function of another variable, ...
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25 views

Matrix representation of quadratic partial differential equations

For a particular problem, I have two following quadratic differential equations: $f_{uu}g_{u} - g_{uu}f_{u}$ = 0 $(f_{uu}g_{v} - g_{uu}f_{v}) + 2(f_{uv}g_{u} - g_{uv}f_{v})$ = 0 here, $f$ and $g$ ...
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33 views

Newton's differential equation

As we all know one of Issac Newton's many achievement was to use his theory of gravitation and his law of motion to determine the way the planets move. I am looking for a not too deep resource in ...
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39 views

Different ways to derive differential equations for sech/csch/sec/csc

I recently realised that I only know one way to derive the differential equations for the reciprocals trigs (the hyperbolic and normal secant and cosecant): brute search with taking derivatives. As ...
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65 views

Problem with commutator relations

part a) is fine. part b) is not. A commutator is defined as, for operators $A$ and $B$, $[A,B]=AB-BA$. [SOLVED]I get that $H(\lambda)=e^{-\lambda D}Ce^{\lambda D}$, $H'(\lambda)=-De^{-\lambda ...
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21 views

Monodromy matrix

I have a periodic difference operator $$(L\psi)_{n}=\psi_{n+k}+a_{n}^{k-1}\psi_{n+k-1}+\ldots+a_{n}^{1}\psi_{n+1}$$ I know how to write monodromy matrix for second order operator.I want to know how ...
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87 views

regular singular point at infinity

Consider the differential equation $xf''(x)+(a+1-x)f'(x)+nf(x)=0$, prove that the differential equation has only two singularities $x=0, \infty$, both are regular. For $x=0$, $$x \frac{a+1-x}{x}, x^2 ...
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61 views

Stiff differential equations without using Jacobian matrix

I want to solve a stiff system of differential equations. Its Jacobian matrix isn't constant and its determinant is close to zero so I cant inverse of it. Please tell me does exist a method that solve ...
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39 views

Critical points of hamiltonian system

Consider the following system: $x'=4y+y(x^2+y^2)$ $y'=4x-x(x^2+y^2)$ Classify all critical points of the system. I know the critical points are $(0,0), (2,0)$ and $(-2,0)$ but im not sure how to ...
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vector field with exactly three tangency points

How can I find equation of a diacritical vector field with exactly three tangency points to the exceptional divisor??? Does anybody have a hint for this? Thanks!
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74 views

Errors in numericaly solving hyperbolic PDE in matlab

I am a beginner for PDE and I want to solve a hyperbolic PDE using matlab's builtin function hyperbolic(). However I am facing some erros and I could not resolve them. Can someone suggest or comment ...
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31 views

how to determine lyaponov functions

i was wondereing if there is one kind of technique determine lyapunov functions. I know for example that if you got $\dot{x}=y$ $\dot{y}=x-x^{3}$ that you can say $L = \frac{1}{2}y^{2} + V(x)$ with ...
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31 views

Wronskian to get particular answer in differential equation

I have the equation $\frac{d^2y}{dt^2} + y = \sin(t)e^t$ I have found the corresponding equation which is $A\sin(t)+B\cos(t)$ where $A$ and $B$ are coefficients I am now working out the particular ...
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24 views

“Differential variations”?

This passage in an old book on trigonometry calls these relations among parts of a spherical triangle "differential variations". The "parts" are three sides and the three angles; when the sides are ...
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52 views

Maximum and Minimum principle of Laplace equation

I want to prove that if $u=u(x, y, t)$ is a solution of the equation: $$\frac{\partial u}{\partial t} = \Delta u,\;\;\;\;\;\;\ where \; \Delta u = \frac{\partial ^2}{\partial x^2} + \frac{\partial ...
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44 views

The existence and uniqueness theorem

I'm struggling to understand how and when to use this theorem to solve problems, and was wondering if I could get any tips here. $y'=-y-2$ $, y(0)=0$ Determine $ø_n(t)$ for an arbitrary value of ...
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88 views

How to eliminate functions from a set of differential equations with Maple?

I've got a set of 7 equations, 3 of them being differential equations, with 7 unknown functions. I'd like to reduce the system to a set of 2 differential equations by eliminating 5 of the unknown ...
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25 views

ODE in case of discontinuity

Consider a function $\varphi(x) : \mathbb{R} \rightarrow \mathbb{R}$ bounded and continuous and $c \in \mathbb{R}$. Question 1 Is there a unique $f : \mathbb{R} \rightarrow \mathbb{R}$, among all ...
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273 views

How to represent Non-linear equation in State Space form? (To solve in MATLAB)

I have a set of differential equations. I am using state space representation to convert it 1st order form and then am solving via RK method using ode45 function. I know how to do this when the ...
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Impulsive Boundary value problems

I have this paper They consider this impulsive problem i dont understand this : Proof. First, suppose that $x\in E\cap C^2[J',R]$ is a solution of problem $(1.5)$. It is easy to see by ...
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42 views

What Laplace transformation is used for this?

I cannot see the transition here. I was unable to find what Laplace transformation was used here.
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42 views

Unable to understand certain mathematical assumptions: Differential Equations

I was looking at page 49 of http://www.macs.hw.ac.uk/~bernd/F13YB1/odenotes5.pdf (page 2 of the PDF) And I came across a relatively strange argument which didn't make logical sense to me: The ...
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86 views

5 Parameter Affine Transformation

I am working on computing affine transformation using Gradient Ascent Method, so the Inverse compositional algorithm. However, I am stuck in one probably simple step but I fail to understand them. ...
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28 views

Second-order Euler Cauchy; regarding choice of $\lambda$

The exact equation is: $$\quad y'' - \dfrac{4}{x}y' + \dfrac{6}{x^2}y = 0, \quad x > 0.$$ So, solving it was okay, but I literally always stumble upon the same dilemma. What do I choose to be ...
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17 views

Collocation method

Can anyone solve the boundary value problem by Collocation method step by step for given boundary conditions? $Y"= 0$ considering $0 \le x \le 2$ given that: $h = 0$ when $x = 0$ and $h = 10$ when $x ...
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43 views

Find the solution of this differential equation

I want to solve $\dot{\xi}(s)=\sqrt{\frac{(n-2)^2}{4}\xi(s)^2-\frac{n-2}{n}\xi(s)^{\frac{2n}{n-2}}}$ with the condition $\xi(0)=\biggl(\frac{n(n-2)}{4}\biggl)^{\frac{n-2}{4}}$. I know that ...
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25 views

Second Level Operators:

What would be an example of an Operator $$H$$ such that for any and all explicit functions U $$H[u] = I$$ where I is some other function However, for some other Operator W ex: [d/dx] ...
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26 views

Find the values ​​of $\alpha$ and $\beta$ for which $x' = at^{\alpha}+bx^{\beta}$ becomes an equation homogenously

Find the values ​​of $\alpha$ and $\beta$ for which $$x' = at^{\alpha}+bx^{\beta}$$ becomes an equation homogenously through a change of variables of the form $x = y^m$
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Dynamic Pari-Mutuel Markets, generalizing to multiple discrete outcomes

In the paper http://dpennock.com/papers/pennock-ec-2004-dynamic-parimutuel.pdf the author describes a mathematical method creating a dynamic pari-mutuel market with an automated market maker. The ...
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what does well posdeness results tells us concerning non linear evolution equations?

Consider a nonlinear Shr\"odinger equation, $$iu_{t}+\bigtriangleup u + f(u)= 0, u(0)= u_{0}$$ where $u(t, x)$ is complex valued function of $(t,x) \in \mathbb R \times \mathbb R^{n}$, $i=\sqrt{-1}, ...
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151 views

What do Root[], #, & mean in Wolfram Alpha?

I wanted to find the roots of a particular equation by using Wolfram Alpha. What I get are strange answers. Thay look like this: What do all the symbols there mean and what does the whole thing mean? ...
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PDE Heat Equation Question: Finding T(x,t) with limited information.

Say our equation for temperature at position x and time t is shown by: $$ T(x)=T_0(1-x/a) $$ This equation holds for a rod of length a from x=0 to x=a. Initially T(0,t)=$T_0$ and T(a,t)=0. ...
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Is there a solution for this stochastic differential equation or analogous ordinary differential equation?

I'm trying to analyze the following Ito stochastic differential equation: $$dX_t = \|X_t\|dW_t$$ where $X_t, dX_t, W_t, dW_t \in \mathbb{R}^n$. Here, $dW_t$ is the standard Wiener process and ...
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35 views

Solving $u_{yy} + (2-x)u_y - 2xu = 1$

I want to solve the pde $$ u_{yy} + (2-x)u_y - 2xu = 1 $$ so if I treat $x$ in the coefficients as arbitrary but fixed it is equivalent to solving the ode $$ y'' + (2-x) y' - 2x y = 1. $$ For the ...
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99 views

Jacobi and Gauss Seidel Iteration for solution of ODEs

I have used the Jacobi and Gauss-Seidel iteration schemes for solution of the following ODE: $$y^{''}(x)-5y^{'}(x)+10y(x)=10x $$ I will outline my method below: Discretion the equation by ...
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88 views

Derivation of Euler Lagrange Equation

I was reading on the derivation of the Euler Lagrange Equations (in the link: http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation focusing on: "Derivation of one-dimensional Euler–Lagrange ...
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21 views

Linear Transformation of Variables

I am wondering if there is some sort of theory/trick that can help me solve this problem: This is for my non-linear dynamics course. We are studying pitchfork bifurcations and the problem is as ...
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98 views

What about uniqueness of general solution?

I found some info about uniqueness for inital value problem. But what about uniqueness of general solution? Is it right that ODE $y'=y$ has two general solutions? 1) $y=Ce^x$ 2) $y=e^{(x+C)}$ Or ...
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44 views

predictor-corrector method and stability

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
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Is there a program for convenient working with equations and coefficients?

I perform some calculations with one differential equation. Then I got a huge expression depending on $x$ and its degrees/powers. E.g. $$\alpha x+(4-x+\sqrt[3]{x})^2-(\beta\sqrt{x}+\frac12(x^3+1))^3 + ...
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advance numerical solution of differential equations

Given the equations for the harmonic oscillator $\frac{dy}{dz}=z, \frac{dz}{dt}= -y$ if the system is approximated by the symplectic Euler method, then it gives $z_{n+1}= z_{n}-hy_{n}, \\ ...
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34 views

Ordinary Differential Equation and linear algebra

Consider the ODE $$-u_{k+1} + 2u_{k} - u_{k-1} = \frac{k}{(n+1)^{3}}$$ With the conditions $u_{0}=0$ and $u_{n+1}=0$, $k=1,...,n$. The points $P_{k} = \left( \frac{k}{n+1}, u_{k}\right)$, $k= ...
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47 views

Sturm-Liouville problems

Suppose that the following Sturm-Liouville problem (for $\lambda\in\mathbb{R}$) for $t\in[0,1]$ $$\begin{cases}-(p(x)\phi'(x))'+q(x)\phi(x)=\lambda\phi(x) \\ \alpha_1\phi'(0)+\beta_1\phi(0)=0 \\ ...
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$\nabla^2f(x,y,z)=0$ in $D$, $\vec{n}\cdot \nabla f=0$ in $ \delta D$, deduce $f=$ constant

Suppose, $$\nabla^2f(x,y,z)=0\quad (x,y,z)\in D,$$ and $$\vec{n}\cdot \nabla f=0\quad (x,y,z)\in \delta D$$ where $\delta D$ is the boundary of some region $D$. How do we deduce that $f=$ constant?
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Is it possible to apply the Melnikov function to nonperiodic perturbations?

In the case of planar Hamiltonian system, the classical Melnikov function deals with the periodic perturbation. Is it possible to apply the Melnikov function to nonperiodic perturbations?
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40 views

Integral-Differential Equation Modeling Banked Turn

Solve this equation for the function $y(x)$: $y' = \alpha \left(\int\sqrt{1 + y'^2} dx \right)^2$ Of course this must first be solved for $y'$ and then integrated to get $y$. The following is not ...
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34 views

Solving non-autonomous ODE $y' = 2 | x | y$

So, a past-years' test asked students to find the general solution of $y' = 2 | x | y$, and, after that, to find the solutions with initial conditions $y(0) = 1$ and $y(-1) = -1$. First, this ...
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Find det M in the differential equation

Let's take a scalar equation: $\left(\tan{t}\right)y'''+2y''-\left(\sin{t}\right)y'+2y=0$ Defined on the interval $ t \in (-\frac{\pi}{2}, \frac{\pi}{2})$ Let $M(t,\frac{\pi}{4})$ be a solving matrix ...