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

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1answer
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Poincare-Bendixson theorem contradiction help

Lets suppose p is asymptotically stable but not a singularity for the planar differential equation dx/dt=f(x), then for points x sufficiently closed to p we must have x(t) tends to p. so the limit set ...
0
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1answer
47 views

Theorem with an example

I have this theorem In the paper they give an example: But here $H_1$ is not satisfied ! How to correct it please? http://mathoverflow.net/questions/163788/theorem-with-an-example
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0answers
21 views

near identity change of coordinates

Problem: Consider the scalar differential equation $$x' = \frac{4x – 24x^2 – 16x^3}{1 – 12x – 12x^2}.$$ which has a fixed point at $x^* = 0 $. For $x$ close to $x^* = 0 $ find a near identity ...
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1answer
24 views

Heat Equation Steady state question

Say you have a slab of material occupying the region $0\leq x\leq a$. Heat is supplied at a constant unit rate so the temperature T(x,t) satisfies $\partial T$/$\partial t$= $k$ $\partial^2 ...
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2answers
64 views

Solve second order differential equation with Heaviside function using Laplace transform

The equation is: $$y'' + 3y = u_4(t)\cos(5(t-4)), \quad y(0) = 0, \quad y'(0) = -2$$ Here $u_4$ is the Heaviside function with activation switch at $t=4$. I can get all the way to the partial ...
1
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1answer
12 views

How to solve an ODE with boundary conditions using Matlab solver?

My question is very simple: I want to plot a graphic for the deflection of a beam, with consists of a solution of an ODE using a Matlab solver, such as: %Call Solver -> Linear [x y] = ...
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0answers
5 views

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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2answers
56 views

Why this ODE doesn't have a solution?

Consider the following problem: $$u'' + u = \sin t ,\,\, 0 < t < \pi$$ $$u(0) = u(\pi)=0 $$ My book says that this problem doesn't have a solution (classic solution). I don't see how to ...
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2answers
53 views

Does a solution to the differential equation $y'=y$ exist?

What is the solution to this differential equation : $$f'(x)=f(x)$$ I'm very interested in this because if it have a solution this means that the slope of that function at a point $a_0$ is the height ...
2
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0answers
15 views

Existence and uniqueness of initial value problem in differential equation

consider the following equation: $$ y'=y^{\frac{1}{3}}, \,y(0)=0 $$ My question is how can I prove the existence and uniqueness of solutions of this initial value problem without solving the ...
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0answers
25 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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0answers
17 views

poincare-bendixson theorem contradiction

Lets suppose p is asymptotically stable but not a singularity for the planar differential equation dx/dt=f(x), then for points x sufficiently closed to p we must have x(t) tends to p. so the limit set ...
3
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0answers
27 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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0answers
22 views

Could explain me the following property of the mean value?

Could explain me the following property of the mean value? Let $u$ be a function that satisfies the Laplace equation at a disc that is continuous at the boundary of the disc. Then the value of $u$ ...
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0answers
16 views

Question about Poisson formula

We have the Laplace equation in polar coordinates: $$u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta \theta}=0, 0 \leq r <a, 0 \leq \theta \leq 2 \pi$$ With the separation of variables, the solution ...
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0answers
12 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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1answer
24 views

Separable Differential Equation

The question is: $$t^5\frac{\mathrm{d}y}{\mathrm{d}t} + y^5 = 0$$ The next step says $\frac{1}{y^5}\frac{\mathrm{d}y}{\mathrm{d}t} + \frac{1}{t^5} = 0$ i understand this. However it then says: ...
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0answers
15 views

Blowup of ODEs in the presence of local Lipschitzianity?

Pardon me if the question is trivial, but I am failing to decide it. Assume that we are given an ODE system $\dot{x} = f(x)$ with positive initial conditions $x(0)$ and know that $f$ is locally ...
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1answer
40 views

Given one solution, can a second solution always be found?

Let's consider a second order ODE: $$y''+p(x)y'+q(x)y=f(x)$$ A common procedure is to find linearly independent solutions $y_1,y_2$ to the homogenous ODE, and then apply the technique of variation ...
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1answer
14 views

Is that the general solution of the Helmholtz equation?

Helmholtz equation: $$u_{xx}+u_{yy}+k^2u=0$$ $$0 \leq x \leq L$$ $$0 \leq y \leq L$$ The solution is in the form $u=X(x)Y(y)$ Replacing this at the equation we get the following problems: ...
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0answers
26 views

Solution to differential equation in symmetric form

Can someone help me to solve this system of differential equations in symmetric form: $$ dx/y*(x+y)= -dy/x*(x+y) = dz/(x-y)(2*x+2*y+z) $$ From 1st and 2nd equation I get $x^2+y^2=C$. I also manage ...
3
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1answer
37 views

Bessel Equations Addition Formula

So, I'm considering yet another tricky proof involving Bessel Functions. Basically, I'm trying to figure out how the following is true: $$J_n(\alpha + \beta) = \sum_{m = -\infty}^\infty ...
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1answer
23 views

Laplace’s equation in the Polar Coordinate System

Laplace’s equation in the Polar Coordinate System: ...
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0answers
23 views

Logistic equation model.

I need some help on the following question: A population of insects increases at a rate r proportional to the total population. Initially, there are 20000 insects, and birds eat 1000 insects per ...
2
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1answer
26 views

Solution in common for two differential equations

Consider: $E1: y''-4y'+4y=0$ Solution: $y(x)=c_1 e^{2x}+c_2 x e^{2x} $ $E2: y''-2ay'+(a^2-1)y=0$ Solution: $y(x)=c_1 e^{(a+1)x}+c_2 e^{(a-1)x} $ For what values of $a$, $E1$ and $E2$ have ...
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0answers
12 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 ...
0
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1answer
25 views

Recommend resources on dynamical systems and singularities

I'm looking for resources on bifurcation theory and systems of non-linear differential equations, but am very particular about the way it is taught/explained. I would like the approach to be based on ...
2
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0answers
39 views

Counterexample to Peano's theorem in infinite dimension

Would you like a counter example that Peano's theorem does not apply to spaces with infinite dimension. Peano theorem: Let E be a space with finite dimension, consider a point $(t_0,x_0) \in \Re ...
2
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0answers
17 views

Uniqueness of the solution to a certain IVP

Let $f:[0,1]\to[0,1]$ be a strictly decreasing, continuous function with $f(0)=1$ and $f(1)=0$, and consider the following IVP: $$\frac{dy}{dt}=f(x(t))-y(t), \ \ \ y(0)=0$$ ...
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0answers
8 views

Differential equation with variable change of function composition

I need to find all the functions $z(x)$ for $z'-e^{x^3} cos(x) z = 3x^2 z L(z)$ As sugerence, i have a proposed variable change: $y(x)=L(z(x))$ To do that var change, i need to express the equation ...
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3answers
49 views

what are the equilibrium points of the following: [on hold]

where $x$ represents susceptible individuals, $y$ represents infected individuals. Find the two biologically meaningful equilibria. $$ \frac{\mathrm{d}x}{\mathrm{d}t} =12−3xy−3x $$ $$ ...
0
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1answer
11 views

Trying to differentte $\ln(|2+f(x)|)=2+e^{x*x}$

I am trying to solve this differential $\ln(|2+f(x)|)=2+e^{x*x}$ so far I did this much; $$ \ln(|2+f(x)|)=2+e^{x*x}\\ |2+f(x)|=e^{2+e^{x*x}}\\ \text{now I have two situations/solutions, because of ...
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1answer
40 views

Bessel Functions Integral Representation proof

So, I'm still working with Bessel functions and trying to proof the following identity, but I'm at a loss for what could possibly be going on here: Any idea how to even approach the proof for ...
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0answers
16 views

Homogenous Linear ODE with constant coefficients

How do you factor the following Homogenous Linear ODE with constant coefficients and what is the general solution: $$L[f] = \left(\frac{\mathrm{d}}{\mathrm{d}x} ...
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0answers
9 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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0answers
21 views

Why do we want that the determinant of the coefficients is $0$?

Eigenvalue problem with periodic boundary conditions-complete Fourier series $$y''+\lambda y=0, 0 \leq x \leq L$$ $$(*): \begin{cases} y(0)=y(L)\\[4pt] y'(0)=y'(L) \end{cases}$$ $$$$ It's a ...
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0answers
28 views

Equilibrium question [on hold]

Consider the differential equation $$x' = x^3 − x^2 − 6x.$$ (a) Find all equilibria. (b) Determine the stability of each equilibrium analytically (not from the phase line diagram). (c) Sketch ...
0
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1answer
12 views

Solving a PDE: basic first-order hyperbolic equation $u_t+cu_x=0$

So I have to solve the first-order hyperbolic equation $u_t+cu_x=0$ and $c$ as a constant. It is a PDE, since there is the time and spatial variable, but I'm overwhelmed by the maths given in books of ...
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1answer
22 views

Need help with proving a lemma

I need to prove the following with the help of Gronwall's inequality: If, for $t \in [a,b]$, $$\phi(t) \leq \delta_2(t-a) + \delta_1 \int_{a}^{t}\phi(s)ds + \delta_3,$$ where $\phi$ is a nonnegative ...
0
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0answers
19 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 ...
1
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1answer
37 views

Need helping proving that something is differentiable but not continuously differentiable

I need some help please proving that a function is differentiable at $(0,0)$ but not continuously differentiable at $(0,0)$. The function is as follows... (from $\mathbb{R}^2$ to $\mathbb{R}$) ...
2
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1answer
28 views

Ordinary Differentiation $t^2y''=t(t+2)y'-(t+2)y$

$$ t^2y''=t(t+2)y'-(t+2)y $$ The question is how to find the Wronskian without knowing the solutions of this equation? I uploaded the origin question below, which is from a sample test. Anyone ...
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1answer
18 views

When does Initial Value Problems have: no solutions, more than one solution, precisely one solution?

I haven't taking Differential Equations for over 2 or 3 years and it escapes my memory how to determine when would an IVP (Initial Value Problem) would have no solutions more than one solution ...
1
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1answer
40 views

Laplace's equation-separation of variables

I am looking at the $2$-D Laplace's equation $$\nabla^2u=u_{xx}+u_{yy}=0$$ $$u(x,0)=f(x), x \in (0,a)$$ $$u(x,b)=0, x \in (0, a)$$ $$u(0,y)=u(a,y)=0, y \in (0,b)$$ The solution is in the form ...
1
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1answer
26 views

Take the Laplace Transform

Take the Laplace transform of $$ \int_{0}^{t}x^2(x-t)^4 \cos(x)dx .$$ I'm not quite sure where to start...
0
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1answer
13 views

Finding zeroes of a numerical solution of an ODE in Maple

I have a system of ODEs involving many variables, say 20, and I have solved this system numerically by Maple for a particular initial condition. When I plotted these solutions it was clear that ...
3
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1answer
23 views

Help solving an ODE

This is an example in my book. It is for the following system: \begin{align*} x'&=y+x(1-x^2-y^2)\\ y'&=-x+y(1-x^2-y^2) \end{align*} So using polar coordinates we get the following system ...
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0answers
32 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 ...
1
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1answer
31 views

How to find I(t)?

I'm working with a SIS model for diseases. Where S stands for susceptibles, and I stands for infected. I have a situation that is modeled by the system: $$S'(t)=\frac{dS}{dt}=-\beta SI-\lambda S$$ ...
0
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1answer
85 views

Best Book For Differential Equations?

I know this is a subjective question, but I need some opinions on a very good book for learning differential equations. Ideally it should have a variety of problems with worked solutions and be ...