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

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1answer
28 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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1answer
16 views

Separable Differential Equations with log in the question

the question is $y \log y-t(\dfrac{dy}{dt}) = 0$ I have separated the question to $4\dfrac{1}{t} dt = \dfrac{1}{y\log y} dy.$ Integrating would give me $log(t) = \log(y\log y) + c$. How do I ...
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1answer
22 views

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 ...
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1answer
39 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
59 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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19 views

poincare-bendixson theorem contradiction [duplicate]

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 ...
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2answers
81 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 ...
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1answer
26 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
29 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 ...
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0answers
7 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
58 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
62 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 ...
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2answers
51 views

Solving a differential equation?

I'm trying to analyze the transient state of a RC circuit. My book gives me the following differential equation: $$\frac{d(v(t))}{dt} + av(t) = c$$ for some constants $a$ and $c$. The book thens ...
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0answers
18 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
34 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
30 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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2answers
68 views

the global stable and unstable manifolds

Show that $x^* = (1, 2)$ is a fixed point of the system $x_1' = 2 + 3x_1 − 2x_2 − x_1^2 + 2x_1x_2 − x_2^2$ $x_2' = 3 + 4x_1 − 3x_2 − x_1^2 + 2x_1x_2 − x_2^2$ Determine $W^s(x)$ and $W^u(x)$, the ...
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1answer
33 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 ...
1
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1answer
55 views

Bessel Functions Proof

How would I even begin to start proving the following? After looking at Frobenius' method and the Rieman P-equation, I started delving into the derivation of Bessel's/Legendre's functions, and I ...
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1answer
25 views

Laplace’s equation in the Polar Coordinate System

Laplace’s equation in the Polar Coordinate System: ...
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2answers
112 views

Integrate $\int^{ln(2)}_0 (3e^u - e^{2u} - 2)\sin(nu)du$

I'm having trouble integrating this function $$\begin{equation} \begin{split} f(x) & = \int^1_0x(1-x)\sqrt{1+x}\sqrt{1+x}\sin(n \ln(1+x))/[(1+x)^2] = \\ & = ...
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1answer
25 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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1answer
43 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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0answers
16 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
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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2answers
140 views

Real analysis question involving inhomogenous linear ODE

So I had another problem like this but the ODE was homogenous, now there is a non zero right side. I completed part (i), $\large c(x) = \int \frac{b(x)}{g(x)} dx$. I am stuck on (v). (1) is the ...
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2answers
43 views

Solving a certain differential equation when assuming a surface of revolution is minimal

The problem is the following: Consider the surface of revolution $$ \textbf{q} (t, \mu) = (r(t)\cos(\mu),r(t)\sin(\mu),t) $$ If $\textbf{q}$ is minimal, then $r(t) = a\cosh(t)+b\sinh(t)$ for $a,b$ ...
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3answers
63 views

what are the equilibrium points of the following: [closed]

where $x$ represents susceptible individuals, $y$ represents infected individuals. Find the two biologically meaningful equilibria. $$ \frac{\mathrm{d}x}{\mathrm{d}t} =12−3xy−3x $$ $$ ...
3
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1answer
50 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
40 views

How to show no periodic orbits exist

I am trying to show that no periodic orbits exist for the system: $$ x_1'=y+x^2+xy^3$$ $$y'=-2x-y^3$$ I have tried using Dulac's criterion to find a function $g(x,y)$ such that $\Phi(x,y)$ given by ...
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1answer
16 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 ...
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0answers
28 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 ...
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1answer
28 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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13 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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0answers
40 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 ...
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1answer
41 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 ...
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0answers
18 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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1answer
36 views

How can I solve an ODE when $F(x_0)=F'(x_0)=0$ is given at an unknown point $x=x_0$ using bvp5c?

I'm attempting to solve the following ODE using MATLAB bvp5c. I've used bvp5c for other typical multipoint boundary value problems but I have no idea how to deal with ODEs with conditions given at an ...
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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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2answers
67 views

linear differential equation problem [closed]

Consider the following system of linear differential equations: $$\begin{split} \frac{dx}{dt}&=−3x+y\\ \frac{dy}{dt}&=x−3y \end{split}$$ Find the eigenvalues and eigenvectors associated ...
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1answer
55 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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1answer
49 views

Differential equation question

Consider the differential equation $\dfrac{dx}{dt}=x^3−x^2−6x$ . Find all equilibria. Determine the stability of each equilibrium analytically (not from the phase line diagram). Sketch the ...
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1answer
23 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 ...
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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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0answers
16 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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28 views

Equilibrium question [closed]

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 ...
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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 ...
2
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0answers
60 views

second order linear ode in the complex domain

Consider $w''(z)+p(z)w'(z)+q(z)=0$ where $p(z), q(z)$ are analytic for $R\le|z|<\infty$ for some fixed $R$. Now I want to prove using analytic continuation of the solutions that the ode has one ...
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0answers
20 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 ...