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

learn more… | top users | synonyms (1)

0
votes
0answers
10 views

differential equation as taylor series

Consider the equation $\frac{d x(t)}{dt} = g(x(t))$ , with $x(0) = x_0$, where g is function that admits derivatives of all orders.If the solution of the equation can be written as a series of taylor ...
0
votes
0answers
4 views

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}, ...
0
votes
0answers
7 views

limit set of a bounded solution

Going over my lecture notes I have noticed that my professor uses poincare bendixson theorem whenever we have a solution x(t) that remain bounded for all time, but shouldn't we also know that the ...
1
vote
1answer
15 views

Finding the Asymptotic Curves of a Given Surface

I have to find the asymptotic curves of the surface given by $z = a(\frac{x}{y} + \frac{y}{x})$, for constant $a \neq 0$. I guess that what was meant by that statement is that surface $S$ can be ...
1
vote
2answers
43 views

Show that there is a unique matrix $ A $ such that $ \varphi (t) = e ^ {tA} $.

Let $ \varphi:\mathbb R\to\mathcal M_{n\times n}(\mathbb R)$ be a function from $\mathbb R$ to the space of $n\times n$ real-valued matrices, and suppose that each component of $\varphi$ is a $C^1$ ...
0
votes
0answers
17 views

List all equations for straight line! [on hold]

Can someone list all the equations for a straight line geometry? Thank You.
0
votes
0answers
19 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? ...
2
votes
2answers
24 views

Linear Differential Equation achieving the answer.

The question states $t(\dfrac{dy}{dt}) - 3y = t^4$ As a first step I am told to divide through by $t^4$ - can anyone explain the purpose of this? Following this I get $t^{-3} (\dfrac{dy}{dt}) - ...
0
votes
0answers
26 views

True or false: differentiation. [on hold]

If the function $f(x,y): \mathbb{R}^2 \longrightarrow \mathbb{R}^3$ is differentiable at $(2,-1)$ with a tangent plane such as $z= 2x - 3y + 2$, then the function $g(x,y)= 3x - 2f(x,y) + 5$ is ...
2
votes
1answer
27 views

Let $p(t)$ be a polynomial in $\mathbb{R}$. Defining $p_0(t) = p(t)$ [on hold]

Let $p(t)$ be a polynomial in $\mathbb{R}$. Defining $p_0(t) = p(t), p_1(t) = 1 + \int_0^t p_0(s)ds, ... , p_k = 1+ \int_0^t p_{k-1}(s)ds.$ Prove that $p_k(t)$ converges uniformly on each compact ...
1
vote
0answers
18 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 ...
1
vote
1answer
401 views

Legendre polynomials recurrence relation

How can i get? $$P_{n+1}=xP_n(x)-\frac{1-x^2}{n+1} P'_n(x)$$ $n>=0$ Also know as the leadder equation of the legendre polinomials i tried to use de recurrence relations as: ...
0
votes
0answers
25 views

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. ...
11
votes
1answer
113 views

Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''\,dx=\int_0^ty'y''\,dx$.

Let $y(x)$ be a solution to $y''+e^xy=0$. Prove that there exists $t$ such that $0\le t\le T$ and $$\int_0^Te^{-x} y'y'' \, dx=\int_0^ty'y''\,dx.$$
1
vote
2answers
184 views

Comparison theorem for systems of ODE

Let vector-function $x(t)$ satisfy a differential equation $$ \dot x = f(x), $$ and a vector-function $y($t) satisfy a differential inequality $$ \dot y \leq f(y) $$ with starting positions $y(0) ...
2
votes
1answer
40 views

Let $A$ a matrix with real or complex entries. Proof that $\displaystyle\lim_{n\rightarrow\infty}(E+\frac{A}{n})^n=e^A, E=$indentity.

Let $A$ a matrix with real or complex entries. Proof that $\displaystyle\lim_{n\rightarrow\infty}\left(E+\frac{A}{n}\right)^n=e^A, E=$indentity. I thought of using the limit, but do not know where ...
1
vote
1answer
26 views

Suppose $\mu$ is not an eigenvalue of A. Show that the equation $x'= Ax + e^{\mu t}b$.

Suppose $\mu$ is not an eigenvalue of $A$. Show that the equation $x'= Ax + e^{\mu t}b$ has a solution of the form $\varphi(t) = ve^{\mu t}$.
1
vote
2answers
16 views

Laplace Transform of tsin(at) using only the definition

Hello I' am stuck on how to get the final result of the laplace transform of $f(t)=tsin(at)$using (a is a constant) only the definition of $$\int_0^{\infty}f(t)e^{-st}dt$$, I know $sin(at)= {1 \over ...
1
vote
1answer
22 views

Show a 2-form is exact finding a primitive.

I have to show that $\omega=-4xy\:\mathrm{d}x\wedge \mathrm{d}y-2xz\:\mathrm{d}z\wedge \mathrm{d}x +2yz\:\mathrm{d}y\wedge \mathrm{d}z$ is exact finding a primitve of $\omega$ (by Poincare lemma I ...
1
vote
1answer
22 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 ...
0
votes
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 ...
0
votes
1answer
20 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 ...
0
votes
1answer
33 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 ...
0
votes
1answer
57 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
0
votes
0answers
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 ...
1
vote
2answers
68 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 ...
0
votes
1answer
32 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 ...
1
vote
1answer
18 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] = ...
0
votes
0answers
27 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 ...
0
votes
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 ...
0
votes
2answers
57 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 ...
1
vote
2answers
57 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 ...
1
vote
2answers
50 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 ...
2
votes
0answers
17 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 ...
3
votes
0answers
30 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: ...
0
votes
0answers
28 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 ...
2
votes
2answers
66 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 ...
0
votes
1answer
31 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
vote
1answer
49 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 ...
0
votes
1answer
23 views

Laplace’s equation in the Polar Coordinate System

Laplace’s equation in the Polar Coordinate System: ...
1
vote
2answers
109 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] = \\ & = ...
0
votes
0answers
24 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 $u$ ...
0
votes
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: ...
0
votes
1answer
41 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 ...
0
votes
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 ...
0
votes
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: ...
1
vote
2answers
132 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 ...
1
vote
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 $a,b$ ...
0
votes
3answers
50 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. $$ \frac{\mathrm{d}x}{\mathrm{d}t} =12−3xy−3x $$ $$ ...
3
votes
1answer
42 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 ...