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

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Change MATLAB code from Lax-Wendroff to Leapfrog

I want to see how leapfrog would look using this code, but I'm having issues implementing it. I think my biggest problem is adding in the $ U_j^{n-1}$ term, I just don't get the logic. Here's what ...
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51 views

Solving ODE numerically - getting local truncation error

Well I have NO idea how to do this or even where to start Compute the order of magnitude of the local truncation error of the following time integration scheme: $$y_{n+1} = y_{n-1} + 2h f(y_n)$$ H ...
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259 views

Decoupling system of two partial differential equations

If I have the following systems of PDE $$ u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)} = 0,\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)} = 0, $$ where ...
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Closed form for a sequence defined recursively

Let $a_k$ be a sequence such that $a_0=0, a_1=0, a_2=1, a_3=1$ and $$a_{k+4}=-\frac{a_{k}+ka_{k+2}}{(k+1)(k+2)}$$ for $k\ge 0$. My question is: Is a closed form formula for $a_n, n\ge 4$ possible? ...
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25 views

If $f(x) = Ax$, show that for all $t \in \mathbb{R}$, the extreme $x_n = x_n(t)$ of polygon converges to $e^{At} x_0$.

Let $f$ is a vector field in $\mathbb{R}^n$, $x_0 \in \mathbb{R}^n$ and $x_{k+1} = x_k + f(x_k)\Delta t$, $k= 0,1,...,n-1$, where $\Delta t = \frac{t}{n}$. A polygon whose points are the $x_i$ ...
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49 views

Matlab functions of variables

So I am writing a function to compute the following equations for an SIR model: So here's my code: ...
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102 views

Transient Behaviour Transient Property Lorenz Equation

Was reading Lorenz paper "Deterministic Nonperiodic Flow" and it says that if a trajectory is not a fixed point, periodic orbit or quasi-periodic orbit and no transient behaviour then it is a ...
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65 views

Modification of Gronwall's Lemma

Exercise 2.3 in this book: ...
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128 views

Proof of Gronwall's Inequality

I have a question about the proof of Gronwall's inequality as given in Chicone: Ordinary Differential Equations with Applications. Gronwall: Suppose that $a<b$ and let $\alpha, \phi,$ and $\psi$ ...
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59 views

Question about an eigenvalue problem

I have a question... How can I show that the eigenvalue problem $$y''+λy=0$$ $$y(0)=0,$$ $$ y'(0)=\frac{y'(1)}{2}$$ is NOT a Sturm-Liouville problem?
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61 views

Help Finding Critical Points of a Cubic (with 2 parameters)

I am trying to find bifurcation points in 1 dimension, but am having trouble finding critical points of $x'=\mu x -2x^2-x^3+ \delta$ ( where $x$ is my variable, $\mu$ is a parameter, and $\delta$ ...
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124 views

Runge Kutta stability region for forward euler and explicit midpoint

The interval of absolute stability is the intersection of the region of absolute stability in the complex plane with the real axis.Show that Runge Kutta forward Euler and RK explicit midpoint have the ...
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63 views

How can i find the next approximate value with this iteration formula?

My ODE is : $$y'' + 2t(y')^2 = 0 $$ with initial values $$y(0)=2,y'(0)=1$$ and the analytical solution is $$y(t)=\tan^{-1}(t)+2 $$ which we convert to a system $$y_1' = y_2 \\ y_2'=-2t(y_2)^2$$ ...
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156 views

Fourth order Runge-Kutta method validity

I wonder whether the fourth-order Runge-Kutta method is suitable for a second-order linear ODE with dissipative terms modelling free fall of an object through a viscous medium under the act of ...
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143 views

Total differentiation

For each of the functions below use the total diferential to approximate the change in $Y$ due to the given changes in $X$ and $Z$: $Y= X^2 + 4X -Z^2 -2XZ$, where $X=1$ and $Z = 4$ , and $\Delta X=2$ ...
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161 views

Predictor-Corrector for Adams-Moulton

What is the order of the corrector of Adams-Moulton type required in order to apply Milne's method for estimating the error in PECE mode? Find the coefficient of the leading term in the truncation ...
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35 views

Find a solution to $x''+x=g(t)$, $x(t_0)=x_0$, $x'(t_0)=x_0'$

Find a solution to $x''+x=g(t)$, $x(t_0)=x_0$, $x'(t_0)=x_0'$, where $g$ is a continuous function in $\mathbb{R}$. Somebody can give me a hint?
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solving 1st order differential equation.

$$ \frac{\dot{c_t}}{c_t}=f'(k_t)-\delta-\sigma $$ $$ \dot k_t =f(k_t)-c_t-(n+\delta)k_t$$ $$(\delta,\,\sigma \text{ and } n \text{ are parameters}, \, c_t=c(t),\, k_t=k(t)) $$ How can I solve this ...
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32 views

Need to interpret a system of first order linear differential equations as an application to Linear Algebra.

I understand that the Wronskian determines whether solutions are linearly independent. But suppose I have a system like $Ax = x'$, where $x$ is a vector valued function of $t$. Suppose $n$ is the ...
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35 views

Help with semilinear PDE problem

I need some help: Let $T>1$, $\Omega_T=\{(x,y)\in\mathbb{R}\>|\>x\in(1,T)\}$, $\Gamma=\{(x,y)\in\mathbb{R}\>|\>x=1\}$ and $g\in C^1(\mathbb{R})$. Prove that there exists $T>1$ such ...
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91 views

Show that $y/x$ tends to a finite limit as $x \to + \infty$ and determine this limit.

Let $y=f(x)$ be that solution of the differential equation $$y' = \frac{2y^2+x}{3y^2+5}$$ which satisfies the initial condition $f(0)=0$. (Do not attempt to solve this differential equation.) (a) ...
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44 views

What is the explanation of the equation 1-4.6 in the book “Applied Exterior Calculus”?

If the $n$ function $\{f^i(x^m)\}$ are of class $C^{\infty}$ in some neighborhood of $0$, then system of autonomous ODEs $$\frac{d\bar{x}^i(t)}{dt}=f^i(\bar{x}^m(t)),\quad\quad i = 1,\cdots, n$$ has a ...
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43 views

The uniqueness of a solution of a system of differential equations

Suppose I have a system of delay differential equations $\frac{d G_1(\phi_1(s))}{ds}= F_1(\phi_1(s),\phi_2(1-s),1-s)$ $\frac{d G_2(\phi_2(1-s))}{ds}= F_2(\phi_1(s),\phi_2(1-s),s)$ where $G_1, G_2, ...
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32 views

Linear gradient equation in the plane

Observe the following equation: $V(y)=\frac{a}{2}y_1^2+\frac{b}{2}y_2^2+c y_1y_2$ a) Find the matrix $A$ which you find on the right hand side of the equation. b) Calculate the Trace, determinant, ...
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33 views

mathieu function of non integer order asymptotics

I have an asymptotic expression for the integer mathieu functions: $se_\nu(q,z)$ and $ce_\nu(q,z)$, where $\nu$ is an integer. I would like to use these expression for the case $\nu$ real. My question ...
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26 views

Easy ode question.

I forget ODE. Please solve this. Thanks. $M=M(u,v))$ $M_{uu}+M=0$ I guess I need to take $D=\frac{d M}{d u}$ Then I need to write $D^2+1=0$ But I cannot remember properly.
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Flux and trajectories through vector field

I have here a very simple vector field $F(u, v) = 2\pi \binom{a}{b}$ where a and b are fix and $a \neq 0$. I have a parametrization of the surface of the Torus T with $\Phi(u, v) =$ $$ ...
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Assistance solving $x'(t)=t-x(t)^2$

I'm taking a second level ODE class and for part of some problem I need to solve a nonlinear first-order differential equation, but I've never worked with nonlinear problems before (there was no ...
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25 views

joint density random variables with a set of equations

There are $n$ equations: $f_i(x_1,x_2,...,x_n,e_i)=0$, $i$ from $1$ to $n$, where $e_i$ are independent random variables whose expectations are all $0$. $x_i$ are random variables. Suppose the map $e ...
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72 views

Historical study of dynamical system

I am currently doing a historical study on my school project 'study of ODE' which slowly shift to the study of dynamical system as I am interested in pursuing my study of ode from linear system, phase ...
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113 views

Linearization of Implicit ODE (Equations of Motion)

let's say we have a system with vector $q_{(t)}$ representing the degrees of freedom (DoF), and state vector $ x_{(t)} = \left \{ \begin{array}{c c} q_{(t)} \\ \dot{q_{(t)}} \end{array} \right \}$ ...
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77 views

Number of zeros of Wronskian

Is there some relation between the number of zeros of a Wronskian and properties of given functions? Having Wronskian (e.g. $2$ x $2$) $$W(x)=\left|\begin{array}{c}f_1(x) & f_2(x)\\f'_1(x) & ...
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82 views

Lanchester's war model optimization.

Suppose the Lanchester's war model: $f'(t)=-0.5g(t)+x\sin^2(t)$ $g'(t)=-0.5f(t)+\cos^2(t)$ with $f(0)=g(0)=2$. How to estimate how small $x$ can be in order to make $f(t)$ won't reach $0$ on the ...
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98 views

Help with Gronwall's Inequality

I'm trying to solve an extra credit hw problem that has to do with Gronwall's Inequality. I understand (or think I do) how to find an upper bound when $p=0$, but I'm unsure of how to handle the extra ...
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57 views

Lyapunov stability of all solutions

Suppose that all solutions to the homogeneous linear differential system $X'(t)=A(t)X$ is bounded on $t\in [0,\infty)$. Show that all solutions are Lyapunov stable.
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strong maximum principle for $u$ such that $u'' \geq c(x)u$

I want to prove that if $I$ is an interval in $R$, $c(x) \geq 0$ is continuous and $u \leq 0$ is $C^2$ then if for all $x$ $$u''(x) \geq c(x)u(x)$$ then the strong maximum principle holds for $u$, ...
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Sensitivity of coefficients in ODE

I am trying to formulate a mathematical model as part of an op-research problem, and I'm running into a roadblock concerning differential equations of a certain kind; I was hoping to understand if ...
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20 views

A two-point boundary problem (ODE)

Would you know if the following ODE admit analytical solutions: $F(x)F''(x)=D(x)$ with $x\in (0,1), F(0)=0, F'(1)=0$ and where D is a continuous, negative function such that $D(0)=D(1)=0$? Does it ...
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382 views

Method of dominant balance and perturbation theory

We know perturbation theory express the desired solution of differential equations in terms of a formal power series in some "small" perturbation parameters: $y=y_0+\epsilon ^1 y_1+\epsilon ^2 ...
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Incomplete solution of a problem in Qualitative ODE.

I can't to complete my solution in the following problem: Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be $C^1$ and $q\in\mathbb{R}$ such that $f(q) = 0$ and $f'(q) >0$. Consider the Cauchy ...
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What minimal value of parameter will give nontrivial solution to the equation?

The second order ODE is given: $y''(x) + (\lambda - x^2) y(x) = 0$ with boundary conditions $y(0) = y(1) = 0$ The question is: what minimal value of $\lambda$ will make it have nontrivial solution?
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179 views

Green function and Fourier series

I would like to understand what is wrong with the following computation. Indeed the result appears rather trivial. I have the following differential equation $$ y''(t,t')+z(t)y(t,t')=\delta(t-t') ...
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81 views

What does the solution guess for a PDE mean? - specific question

I got a partial differential equation, where a suggested solution is: $J(W,t)=\dfrac{g(t)^{\gamma}W^{1-\gamma}}{1-\gamma}$ Everything but $g(t)^{\gamma}$ is known. $g(t)^{\gamma}$ is being ...
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160 views

Existence of unique periodic solution of ODE

How to prove or disprove that the ODE $$ y'(x) = y( x )^9 + ( 1+\sin( x)) y(x) +\cos(x) =0 $$ has the unique periodic solution? PS. Its fieldplot done with Maple
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Bifurcations caused by a single additive term

Motivation: A practical dynamical system is often described by an ODE that has a parameter that controls the "power flow" into the system. When no power flows into the system, nothing interesting ...
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51 views

Find a nonlinear system conjugate the linear system $\overrightarrow x' = \left(\begin{array}{cc} 1 & 2\\0&-4 \end{array}\right) \overrightarrow x$.

A nonlinear system that is topologically conjugate in a neighborhood of its equilibrium point to the linear system $\overrightarrow x' = \left(\begin{array}{cc} 1 & 2\\0&-4 \end{array}\right) ...
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123 views

Are there methods to solve coupled integral and integro-differential equations?

I have one fredholm integral equation $$ y(x)=f(x)+\int_0^1 K_1(x,g(x),t)y(x(t))dt$$ and an integro-differential equation $$ \frac{dg(x)}{dx}=h(x)+\int_0^1 K_2(x,y(x),t)g(x(t))dt$$. Are there any ...
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104 views

Algebraic manipulation of Lyapunov function

I have a problem I would like some feedback on. I have spent 6 hours on it examining various techniques (numerically and analytically). I need to find the values of $k$ for which $x^2+ky^2$ is a ...
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31 views

First integrals and invariants of one-parameter groups

Please, how do I go about showing that the first integrals of the following n-th order differential equation: $$ \frac{d^n u}{dx^n} = H(x, u^{n-1})~~ $$ on $M\subset X \times U ...
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50 views

ODE Problem (Differentiating by function)

I'm trying to solve the following ODE: $x' = (t+x)^2$ I was hinted that $dx/dt = 1/(dt/dx)$, which I assume suggests I should derive $t$ with regard to $x$, I'm not sure how to formally justify it ...