0
votes
1answer
30 views

Question on a derivation regarding the non-linear ODE $x'' = -U'(x)$, $U$ potential

Let $U$ be a potential function, and consider the IVP $$ (*) \quad x'' = -U'(x), \qquad x(t_0) = x_0, \quad x'(t_0) = v_0. $$ We suppose the following: (V) Let $x_0, v_0$ be initial values and let ...
3
votes
3answers
104 views

Partial derivative function definition paradox

I've pondered this question over quite alot and haven't been able to find an answer anywhere. I'm going to ask this question from the standpoint of basic thermodynamics. Let's say I define ...
0
votes
0answers
15 views

Green's function to operator

I would like to understand how one can show that the Green's function in this table is a Green's function to the D'Alembert operator? I refer to the wikipedia page about Green's function
1
vote
0answers
31 views

Function satisfies differential equation.

Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$ G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||} $$ satisfies $$ \Box G(x,t,x_0,t_0) = ...
0
votes
2answers
44 views

How to find the derivative of the flow of an autonomous differential equation with respect to $x$

Ok, may be this is a silly question but consider the following. Let $\dot x=f(x)$ be an autonomous differential equation with $f$ having enough smoothness (Say $C^2$). Let $\xi:\mathbb ...
0
votes
0answers
16 views

Constant solutions of separable ODE

Consider the IWP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 $$ for continuous functions $g : I \to \mathbb R$ and $h : U \to \mathbb R$ on open intervals $I, U$ with $(x_0, y_0) \in I\times ...
3
votes
0answers
39 views

Let $\eta (x)=\int_0^\infty e^{at}\xi(\phi_t(x)) dt$ then $\eta$ is a $C^1$ function

Consider the following problem. Suppose that $a>0, r >0$ and $\xi:\mathbb R \to [o,\infty)$ is a $C^2$ which vanishes in the complement of the interval $(-r,r)$. Also suppose that ...
0
votes
1answer
39 views

Linearizing systems about critical points.

$$\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}\def\l{\lambda}\def\f{\frac{\sqrt{11}}{2}}$$ Find all the critical points of the following systems and derive the linearised system about each ...
1
vote
1answer
20 views

Obtaining Picard Iteratives in a coupled system

Problem: Obtain the first 5 Picard´s Iteratives of the Cauchy problem: $(dx/dt) = y $ with $x(t=0)=0 $ and $(dy/dt)=-\sin(x) $ with $y(t=0)=1 $ The function $x\mapsto \sin(x)$ should be ...
0
votes
0answers
32 views

Find the point implied by intermediate value theorem

Consider a function $f(x)$ such that $f(0)=0$ and $$f'(x) = \frac{T-x}{T-f^{-1}(x)} + \frac{T-x}{S}$$ Then we can see that $f'(0)>1$ and $f'(T)=0$. Find $x$ such that $f'(x)=1$, in terms of the ...
0
votes
0answers
29 views

Square a linear ODE

Assuming that I have a linear ODE without any singularities over the complex numbers $$\sum_{k=0}^{n} g_i(x) y^{(k)}(x)=0.$$ Now I substitute $\sqrt{f}:=y$ into this differential equation and square ...
0
votes
1answer
36 views

Local isometric embedding

Every $n$-dimensional smooth Riemannian manifold admits a local isometric embedding of class $C^1$ into $\mathbb R^{n+1}$ by the Nash-Kuiper theorem. An example by Nadirashvili and Yuan shows that in ...
0
votes
0answers
10 views

How to find error constant, global and local error of ODE integration method

In literature regarding open channel flows I bumped into strange ODE integration methods: the first one: $$y_{i+1}=y_i+\Delta x\cdot\sqrt{f_i\cdot f_{i+1}}$$ the second one $$y_{i+1}=y_i+\Delta ...
0
votes
0answers
26 views

ODE Initial value problem formualtion

If I have the following ODE initial value problem, $$\begin{align} y'(t) &= f(t), \quad t>0, \\ y(0) &= y_0. \end{align}$$ Then I was taught that a solution to the problem is given by ...
1
vote
0answers
32 views

PDE and Taylor's formula

I'm looking to a prove that a function that satisfies the following equations is actually $f(x,t)=x^3 \pm tx$ after changing coordinates. Here are the equations: 1) $\frac{\partial^3 f}{\partial ...
1
vote
0answers
39 views

Proving that a Sturm-Liouville problem is in the limit-point/-circle case

I would like to understand techniques anybody is able to detail to me on how one may actually prove that a particular Sturm-Liouville (S-L) problem, i.e., of the form \begin{equation} ...
0
votes
0answers
54 views

On a differential equation problem of international mathematical competition for university students

I am trying to solve problem 2 of this competition: http://www.imc-math.org.uk/imc2009/imc2009-day2-solutions.pdf I have other thought but i couldn't fill in the detail. Consider the initial value ...
2
votes
2answers
100 views

Asymptote of solution of a differential equation without solving it

Consider the following differential equation (domain $\mathbb{R}$): $$ u(x) = 1 - u'(x) $$ and suppose $u(0) = 0$. How can one prove that $u(x) \to 1$ for $x \to \infty$ without solving the ...
9
votes
4answers
163 views

Solution to $y(x) + y'(x) + y''(x) + y'''(x) + \cdots = 0$

Is there a non-trivial solution to the following differential equation? $$y(x) + y'(x) + y''(x) + y'''(x) + \cdots= 0$$ That is, is there a smooth function $y : \mathbb{R} \to \mathbb{R}$ such that ...
2
votes
3answers
119 views

How to reduce this to Sturm-Liouville form?

I have the ODE $$-(1-x^2) \frac{d^2 f(x)}{dx^2} + x \frac{df(x)}{dx}+g(x)f(x)=\lambda f(x)$$ and I want to reduce it to Sturm-liouville form. The problem is that we don't have $2x$ but just $x$. ...
0
votes
1answer
52 views

Doubt on an ODE problem

Consider the following differential equation $$x'(t) = h(x(t))$$ Consider a function $x(t)$ which satisfies the differential equation for $0 \lt t \leq 1$ and another function $y(t)$ for $0.5 \leq ...
1
vote
1answer
51 views

A basic confusion in the proof of Picard's existence theorem

In the proof of Picard's existence theorem of solution of ODE I don't understand the following step: Once it proves that the limit of uniformly convergent series is a continuous function then it ...
9
votes
3answers
513 views

Eigenvalue problem for ODE with singular coefficients, $-(1-x^2) y'' + py'+qy=\lambda y$

(I did not change anything, I just rewrote the ODE in a simpler form): I started with an ODE (first ODE) : $-(1-x^2)y''(x) +x y'(x) - \left( \alpha x + \gamma x^2 \right) y(x) = \lambda y(x),$ ...
4
votes
1answer
116 views

Function whose gradient is of constant norm

Let $f:\mathbb R^n\rightarrow \mathbb R$ be a smooth function such that $\|\nabla f(x)\|=1$ for all $x\in \mathbb R^n$ and $f(0)=0$. I would like to prove that $f$ is linear. I first looked at the ...
1
vote
0answers
21 views

Initial value problem with intermediate value

The Picard Lindelöf theorem I know always assumes that we specify the value at the left end of the time-interval Picard Lindelöf. Is it true that $x'(t) = f(x(t))$ has a unique solution, in an open ...
0
votes
1answer
35 views

differential equation with random coefficient

I am confused with a problem I encountered at hand, not on how to work on it but rather understanding the problem itself: Let $A(x;\omega)$ be a random field taking values in $[a,b]$ where $a,b < ...
1
vote
1answer
36 views

What smooth functions are solutions of an autonomous ODE?

Let $y$ be a smooth function, say $y : \mathbb{R} \rightarrow \mathbb{R}$. When can we find a continuous map $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $y'=f(y)$ ? Obviously it's not always ...
1
vote
0answers
28 views

About the maximal interval of existence

Let $f:\mathbb R\times \mathbb R^n\longrightarrow\mathbb R^n$ be a continuous function such that there exists some $T\in\mathbb R$ with the following property: $$f(T+t,x)= f(t,x)\;\;\forall ...
1
vote
1answer
88 views

Mathieu differential equation

Given the operator $T (\psi)(x):= \psi''(x)-2q \cos(2x)\psi(x)$ with $T : D(T) \subset L^2[0,2\pi] $ I was wondering: What is the right domain $D(T)$ for this operator if we want to solve the ...
3
votes
1answer
145 views

Derivative of the parameter

I have the equation$\begin{cases} x'(t)=x(t)+y(t) \\y'(t)= \mu y^2(t)+x(t)\end{cases}$ Cauchy problem $\begin{cases} x(0)= 1 + \mu \\y(0)=-2\end{cases}$ . I must calculate $\frac{\partial ...
2
votes
1answer
37 views

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be differentiable and $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$.

The Assignment: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be continuously partial differentiable and let $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$. A ...
2
votes
2answers
29 views

mean value property of derivatives in high dimensions

Let $E$ be a path-connected subset of $\mathbb{R}^n$ and $f$ a differentiable function on $E$. Prove or disprove: for any $x,y\in E$, there exists $z\in E$ such that $f(x)-f(y)=\nabla f(z)\cdot ...
0
votes
1answer
23 views

Let U be open and $f: U \rightarrow \mathbb{R}$ be partial differentiable.

The Assignment: Let $U \subset \mathbb{R}^n$ be open and $f : U \rightarrow \mathbb{R}$ be partial differentiable and let all partial directional derivatives be continous function on $U$. Show ...
2
votes
0answers
72 views

Solution to Schrödinger equation $ \partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t)V(x)f(x,t).$

I want to solve $$ i\partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t-t_0)V(x)f(x,t),$$ for any $V \in C^{\infty}[-1,1]$ and $f: [-1,1] \times \mathbb{R_{\ge 0}} \rightarrow \mathbb{C}$. I would ...
5
votes
0answers
95 views

Solve PDE by getting two ODEs

My goal is to solve this PDE for $f:[-1,1] \times \mathbb{R}_{\ge 0}\rightarrow \mathbb{C}$ $$ \partial_t f(x,t) = -\partial_x^2 f(x,t) + g(t)V(x)f(x,t).$$ I would consider this PDE to be solved if ...
0
votes
1answer
26 views

Is the assumption $y \in C^2$ necessary for the Euler method to be of order $p=1$?

In my Intro to numerical analysis course, we did the following. We stated the initial value problem $\dot{y}=\lambda y+f$, where $f \in C[0,\infty)$, and developed the Euler method. Then proved that ...
3
votes
1answer
27 views

Find interval with function that solves ODE $y'(x)=1+(y(x))^2$

Let $g\in C^1(\mathbb{R})$ with $g'\gt 0$ and $g(0)=0$. Show that for the differential equation $$\begin{cases}y'(x) & = \dfrac{1}{g'(y(x))} \\[8pt] y(0) & = 0 \\\end{cases}$$ there exists ...
2
votes
1answer
53 views

Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ z z' $?

This is a follow-up to Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ \dfrac{z'}{z} $?. It turns out that in that case, \begin{align} \text{$ y = ...
5
votes
1answer
86 views

Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ \dfrac{z'}{z} $?

Can an arbitrary function $ y: \mathbb{R} \to \mathbb{R} $ always be expressed as $ \dfrac{z'}{z} $ for some differentiable function $ z: \mathbb{R} \to \mathbb{R} $, or are additional conditions on $ ...
4
votes
1answer
86 views

Under which conditions a solution of an ODE is analytic function?

If I'm not wrong there is a theorem that says that if the conditions for Picard's theorem are satisfied, for an ode $\dot x=f(x,t)$, then the solution of the ode is as smooth as $f$. I think I'm not ...
0
votes
0answers
34 views

Is the following statement on the stability of the forward Euler method true or false?

My text asks whether the following statement is true or false: The forward Euler method for approximating the solution of $x'=\lambda x$ is stable for all $\lambda \in \mathbb R$ and all step ...
7
votes
1answer
204 views

First-term approximation for singular perturbation of ODE (with two turning points)

I'm reading "Introduction to Perturbation Methods" by Mark Holmes, and I came across an exercise that I don't know how to approach. As I decided to independently read this book, I have no ...
1
vote
1answer
26 views

differential inequality of continuous functions

Let $u:[0,+\infty)\to (0,+\infty)$ be a continuous function such that $\int_0^\infty u(x)dx<\infty$. Suppose there exist $a,b>0$ such that $\frac{du}{dx}\leq u(a+bu)$. Prove that ...
0
votes
1answer
24 views

Notation confusing my understanding of a homework problem

Probably ultra simple, but asking google about notation is non-trivial in a case like this. The text is Oksendal's Stochastic Diff Eq and, very simply, the question is as follows: Let $B_t$ ...
2
votes
0answers
145 views

Solving an infinite non autonomous system of differential equations.

For all $\lambda\in\mathbb{R}$, let $J(\lambda)$ be the infinite matrix where $(J(\lambda))_{nn}=\lambda$, $(J(\lambda))_{n,n+1}=1$ for all $n\in\mathbb{N}$, and all other entries are $0$. This matrix ...
5
votes
1answer
317 views

Generalized Legendre differential equation

In an application I encountered the ODE $$ \left( x^2-1 \right) \frac {{\rm d}^{2}}{{\rm d} x^2} f ( x ) +x \left( \frac {\rm d}{{\rm d}x} f (x) \right) ( 8x^2-7 ) -4 (C+1) f( x ) =0. $$ which is ...
7
votes
5answers
390 views

Solving a separable differential equation

Solve the differential equation: $$y'=\frac{1-y^2}{1-x^2}$$ My book says the solution is: $$y=\frac{x+c}{cx+1},$$ where $c$ is a constant. It's been ten minutes I tried to verify if it was correct ...
3
votes
1answer
56 views

ODE standard form.

I noticed that whenever mathematicians talk about Legendre polynomials they bring the ODE to the form $(1-x^2)f''(x)-2xf'(x)+n(n+1)f(x)=0$. When solving Poisson's equation, this form is not the most ...
1
vote
1answer
29 views

smoothness of harmonic functions confusion in proof

can you explain that last two steps? how that $$na(n)r^{n-1}$$ disappeared in next integral?I noticed the transformation of variables but still not able to figure out properly.
0
votes
1answer
47 views

Problems with a differential equation: $y'=|y|+x^2$

$$y'=|y|+x^2$$ Write the solution to the Cauchy problem with $y(a)=0$ for every $a\in\mathbb{R}$ and tell for what values of $a$ the solution is of class $C^2(\mathbb{R})$. What I tried: The ...