2
votes
1answer
43 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 ...
2
votes
2answers
101 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
47 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 ...
2
votes
0answers
100 views

Find the function whose Taylor series is $\log(x)+\log(x+1)+\log(x+2)+\ldots$

How do I find a function $f$ whose Taylor series is $$\log(x)+\log(x+1)+\log(x+2)+\ldots$$ for some point $x=a$? It would seem that $$\left.\frac{\partial^n}{\partial x^n}f(x) \right|_{x=a} = ...
9
votes
2answers
390 views
+150

Understanding this ODE

(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
109 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
0answers
25 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
35 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
25 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
82 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
110 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
36 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
28 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
66 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
88 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
24 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
25 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
52 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
85 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
30 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
187 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
25 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
110 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
310 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
384 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
28 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 ...
2
votes
2answers
104 views

How to prove that the level sets of this function are closed curves in a specific region.

I need to study the Hamiltonian differential system $$ \begin{align} \dot{x} &= -2ye^{-x^2}\\ \dot{y} &= 2xe^{-x^2}(1-y^2) \end{align}$$ with Hamiltonian function $$ \begin{align} H ...
1
vote
1answer
56 views

Write this ODE without any square roots

Given the function $$u(t):=\sqrt{\sum_{i=0}^n \alpha_i t^{2i}}$$ is it possible to plug this into the ODE $$(t^2-1)u''(t)+tu'(t)(1-8a+8at^2)-4(a+a^2-2at^2+n(-a+2at^2)-C)u(t)=0 $$ such that I get a ...
0
votes
0answers
53 views

Rewrite this ODE

I have the ODE $$(t^2-1)f''(t)+tf'(t)(1-8a+8at^2)-4(a+a^2-2at^2+\phi (-a+2at^2))f(t)= 4\lambda f(t),$$ where $\phi \in \mathbb{R}$ and was wondering whether it is possible to write this ODE in ...
1
vote
1answer
33 views

Sturm Liouville problem with additional term.

Imagine you want to solve an ODE on $[a,b] \subset \mathbb{R}$ $f''(x) + (A(x) + B(x))f(x) = \lambda_n f(x)$, where $A,B$ are some smooth functions and $\lambda_n$ the n-th eigenvalue. Furthermore, ...
4
votes
1answer
88 views

Equivalent of $e^{-x^2}\frac{d^n}{dx^n}e^{x^2}$

Let $f(x)=e^{x^2}$ and write $f^{(n)}(x)=P_n(x)f(x)$ where $P_n$ are polynomials. Then find an equivalent for $P_n(x)$ for every fixed positive $x$. My attempt : $f(x)=\exp(x^2) \quad ...
0
votes
1answer
29 views

uniqueness theorem for ODE system

Let $f:\mathbb R \times \mathbb R^n \to \mathbb R^n$ be a given continuous mapping. Suppose there is a constant $K$ such that $||f(t,x)-f(t,y)|| \le K||x-y||$ for all $t \in \mathbb R, x, y, \in ...
6
votes
1answer
57 views

Differential inequality implies inequality for points at distance pi.

Given a function $f$ with $f+f''\ge 0$, show that $f(x)+f(x+\pi) \ge 0$ for all $x$. Note that for sine and cosine both inequalities become equations. It seems reasonable to look at $f+f''=g$, but ...
1
vote
1answer
53 views

Does bounded and continuous implies Lipschitz?

If a function $f : \mathbb{R} \rightarrow \mathbb{R}$ is integrable, bounded and continuous, is it also Lipschitz continuous?
4
votes
4answers
137 views

A question regarding Frobenious method in ODE

Suppose $b(x),c(x)$ are real functions analytic at 0. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. Suppose $r$ is a double root of $r(r-1)+b_0r+c_0=0$. It is well ...
2
votes
3answers
103 views

differential equation $y''(x)-ay^3(x)+by(x)=0$

Hi I am trying to find a solution $y(x)$ to this non linear differential equation $$ y''(x)-ay^3(x)+by(x)=0. $$ I know a nice solution exists, however how can I go about solving this? I know non ...
1
vote
1answer
22 views

How to prove or is there any reference to “integration-by-parts” formula for difference quotients?

I found this identity in Lawrence C.Evans' book 'Partial Differential Equations' 2ed edition, page293, where $\phi \in C_{c}^{\infty}(V) \ and\ V\subset \subset U$, then ...
6
votes
1answer
133 views

Confluent Heun equation. Reduction to standard form.

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions? $$(1-t^2)u_{tt}-tu_t+4\left[n\beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
15
votes
1answer
464 views

How to make a smart guess for this ODE

I am dealing with a strange problem currently, we have a differential equation $$y(x)^2 = \pm \sqrt{-A \cos(x) - B \cos^2(x)+y'(x)-C},$$ where $C, A$ and $B $ are parameters. (The case that either ...
2
votes
1answer
42 views

Solve this Riccati equation

I want to solve the following Ricatti equation analytically $$y'(\theta) \pm y(\theta)^2= \eta \cos(\theta) + \xi \cos^2(\theta) $$. Does anybody know how to do this cause I failed to do so (by ...
0
votes
1answer
35 views

provide all solutions of $dy-y\,dx=0$ and $\int f(x)\;dx = f(x)$?

$e^x$ is the solution of both equations . the differential operator and integration operator has no effect on $e^x$. So, is there any other function that possesses this property . Or is $e^x$ unique ...
0
votes
1answer
43 views

Solving IVP $y'=t|y|^\alpha, \ y(0)=1$

Intro: This is a follow up to my post Application of Picard-Lindelöf to determine uniqueness of a solution to an IVP, where I am trying to verify that the below IVP has a unique solution in some ...
0
votes
0answers
37 views

Impulsive Boundary value problems

I have this paper They consider this impulsive problem i dont understand this : Proof. First, suppose that $x\in E\cap C^2[J',R]$ is a solution of problem $(1.5)$. It is easy to see by ...