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

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Solution of non-linear Fredholm(Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question but got no response. I rephrase it so that anyone who knows operator theory and integral equations would help me out.....I faced a problem in physics which is a non-linear ...
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22 views

Translated Laplace transform

Is there any way to rewrite the Laplace transform is such a way that that one can apply to an IVP not centred at zero, that is, at some $y^{(n)}(a_n) = b_n$ for $n\in\mathbb{N}$ and $a_n ...
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21 views

Is there a fiber bundle approach to nonlinear oscillations?

I've recently been learning about nonlinear oscillations, and I noticed a seemingly strong connection between how the equations of motion are solved/approximated, and fiber bundles (or vector bundles ...
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Are there any symplectic integration techniques that are A-stable (work on stiff equations)?

The first and second Dahlquist Barriers show that (paraphrasing): Explicit multi-step methods cannot be A-stable and thus are not accurate for stiff equations. Implicit multi-step methods will only ...
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40 views

Estimating limit cycle of ODE system

I'm looking at a system of ODEs: $$\dot{x} = -y - \epsilon^2 x + xy^2$$ $$\dot{y} = x -\epsilon^2 y - x^2$$ After plotting these in Matlab I can see there is a limit cycle very close to the origin ...
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48 views

Solving first order non linear ODE

I am trying to solve the following first order non-linear differential equation: $$ \frac{\partial y}{\partial x} = -\sqrt{\frac{2(\sigma + 1)}{\sigma}} \sqrt{-\frac{1}{2y^{2}} + ...
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43 views

How to solve this non-linear, second order ODE

does anyone know how to solve this ODE? $ yy'' +y' +y =0 $ where y is a function of one real variable.
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23 views

sufficient conditions for finite time of existence of integral curves of a vector field

Let $U\subset \mathbb{R}^2$ open, $\partial U\neq \varnothing$, $V\colon U\rightarrow \mathbb{R}^2$ smooth. Let $c\colon [0,t_{max})\rightarrow U$ be an integral curve of $V$, where $t_{max}$ is the ...
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33 views

How to solve this initial-boundary value problem for a PDE

Consider $$u_{tt}-a^2u_{xx}+u_t+a u_x=0,\quad 0<x<\infty,\quad t>0,(*)$$ where $u_t=\frac{\partial u}{\partial t}$ and etc. It is not so hard to use the method of characteristics to solve it ...
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53 views

How to solve the ODE $2x\frac{dy}{dx}=C(1+(\frac{dy}{dx})^2)^2$?

I am struggling with this ODE I obtained when solving the Euler-Lagrange equation. Can any one help me with solving the ODE $$2x\frac{dy}{dx}=C(1+(\frac{dy}{dx})^2)^2$$ Thanks so much! It comes ...
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Time taken to empty a hemispherical shaped tank

The tank has a radius of $2$m when initially filled and has an outlet of cross section $12$ cm2 Outlet flow as I calculated goes according to the law $V(t)=0.6\sqrt{2gh(t)}$. Having found out the ...
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124 views

How to solve this recurrence relation and solving the power series

Take the following recurrence relation into account: $$ a_{n+2} = \frac{1}{(n+1)(n+2)} \sum_{k=0}^n (s_k - (k+1)a_{k+1})(n-k+1)a_{n-k+1} $$ I know that: $$ s_{2m+1} = \frac{(-1)^m}{(2m+1)!} $$ and ...
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42 views

How to solve $\int_{x}^{x+a} f_X(u) du=e^{-2\lambda_1 x} \int_{x-a}^{x} f_X(u) du$

How to solve equation of the type \begin{align*} \int_{x}^{x+a} f(u) du=e^{-\lambda x} \int_{x-a}^{x} f(u) du \end{align*} we want to solve for $f(x)$ where $\lambda,a$ are some constants. Things I ...
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How to find first-order quasi-linear PDEs form second-order quasi-linear PDE?

Transform $u_{tt} u_{xx}-u^{2}_{tx} + uu_{tt} + 1=0 $ into first-order quasi-linear PDEs. Attempt: $u_{tt}(u_{xx}+u)=(u_{tx}-1)(u_{tx}+1)$ To get $u_{tt} = u_{tx}-1\Rightarrow u_t = u_x ...
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72 views

Differential Equation Direction field

What i want to achieve: I want to plot the direction fields of the following three differential equations: 1. Malthusian growth model: $p'(t)=\lambda*p(t)$ with $\lambda=1$ and $p(t)=t$ 2. Linear ...
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7 views

Convergance of DASPK for a non-linear DAE

I have a system of non-linear DAE and I noticed that the system does not converge if some of the equations are not differentiated. For example, if the control volume equation is represented as this: ...
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23 views

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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7 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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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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26 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: ...
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29 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 ...
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36 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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111 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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32 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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20 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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31 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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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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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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61 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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37 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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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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50 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 ...
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49 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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24 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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33 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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24 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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117 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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55 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 ...
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39 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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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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41 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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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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51 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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17 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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64 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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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 $\mu_y=0$ ...