1
vote
1answer
23 views

differential equation as taylor series

Consider the equation $\frac{\mathrm{d} x(t)}{\mathrm{d}t} = g(x(t))$ , 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 ...
2
votes
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 ...
1
vote
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 $\mathbb{R}$) ...
0
votes
0answers
9 views

Finite difference scheme and its stability

The Finite difference scheme: \begin{equation} y_{n+3}-y_{n+1}= \frac {h}{3}(f_{n}-2f_{n+1}+7f_{n+2}) \end{equation} Deduce that the scheme is convergent and find its interval of absolute stability(if ...
0
votes
0answers
6 views

Modified Symplectic Euler

Simple harmonic motion: $y'= -z $, $z'= f(y)$ and the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ deduce that the coresponding approximate ...
0
votes
0answers
20 views

A predictor-corrector method

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
0
votes
2answers
65 views

Proving a function is Lipschitz continuous

Show that the following function is Lipschitz continuous and find a Lipschitz constant $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{ , } |x-1|\leq\frac{1}{2}\text{ , } ...
1
vote
0answers
24 views

Predictor-Corrector for Adams-Moulton

What is the order of the corrector of Adams-Moulton type required in order to apply Milne's method for estimating the error in PECE mode? Find the coefficient of the leading term in the truncation ...
20
votes
2answers
298 views

When does $(uv)'=u'v'?$ [duplicate]

In any calculus course, one of the first thing we learn is that $(uv)'=u'v+v'u$ rather than the what I've written in the title. This got me wondering: when is this dream product rule true? There are ...
0
votes
0answers
18 views

Showing uniqueness of character identity

How would one show that any complex-valued C1 function satisfying the character identity must be of the form exp(cx) for c complex. Given a function f, it is said to satisfy the character identity if ...
3
votes
0answers
42 views

Differential Equation has a unique solution periodic

Let $A(t)$ continuous and periodic of period $S$ in $\Re$. Suppose $x' = Ax$ has $\varphi \equiv 0$ as the only periodic solution of period $S$. Show that there exists $\delta> 0$ such that for ...
0
votes
1answer
22 views

Understanding the proof of $~M$ invariant set $\Rightarrow$ subtangential condition holds

Problem: I want to understand a proof of the claim given in the title. Suppose we have an initial value problem $\{\dot{x}=f(t,x)~,~x(t_0)=x_0\}$ with continious $f$ and solution $x(t)$. Proof: ...
1
vote
2answers
32 views

Find region for which F(x,y) = (x+y)^2 is Lipschitz in y

As the title says, I need to find such a region. Taking any x, and any y1 and y2 I used the expression |F(x,y1) - F(x,y2)| and plugged in the function respectively for y1 and y2. Now I have to find ...
4
votes
2answers
92 views

What are integrating factors, really?

I can follow the rationale for integrating factors well enough, but they still feel like voodoo to me. Every single description of integrating factors I've seen (and I've seen quite a few, including ...
0
votes
0answers
26 views

Designing a state feedback law for a nonholonomic system

Consider the set \begin{equation*} A_r=\left\{(e_x,e_y,L)\in\mathbb{R}^3:e_x=e_y=0,L(t)=\sqrt{\dfrac{\mu}{p_0^3}}t,t\in\mathbb{R}_{\geq0}\right\} \end{equation*} I have been trying to design a state ...
0
votes
0answers
53 views

Help with Gronwall's Inequality

I'm trying to solve an extra credit hw problem that has to do with Gronwall's Inequality. I understand (or think I do) how to find an upper bound when $p=0$, but I'm unsure of how to handle the extra ...
1
vote
1answer
35 views

Show that if $f\colon\mathbb{R}^2\to\mathbb{R}$ is $C^2$, then any nonempty $\omega$-limit for the equation $x'=\nabla f(x)$ is a critical point.

I'm kind of struggling with an exercise I found in a book about Poincaré-Bendixon theory and I would like some help. The exercise is precisely what I wrote on the title: I have to show that if ...
0
votes
1answer
24 views

Want to show that a solution to some ODE is unique.

So here is my problem, I just found the solution $x(t)=\frac{1}{t^2+1}$ to the following differential equation,$$ \dot{x}=-2tx^2,\;x(0)=1$$ Now I would like to show that my solution is unique and I ...
0
votes
0answers
20 views

How do I measue non linearity of solution of differentional equation?

Is there some well known and useful measures of non-linearity for functions? I know this is rather broad question (more precisely two questions), but I'm stuck with my problem and interested in all ...
2
votes
2answers
105 views

I have a differential equation which solution is periodic. What can I tell about right-hand side of such equation?

I have equation of form $$ \frac{dx}{dt} = f(x), $$ and know and for some initial value $x_0$ its solution is periodic with unknown period. What can I tell about $f(x)$ apart from non-linearity (or ...
1
vote
1answer
32 views

Does $C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})=C^1([0,T];H^{s+1})$?

I have asked this question with other problems, but about this part nobody answers. So I want to ask again, and put some details in it. My question is whether the following equality is correct. If it ...
1
vote
0answers
23 views

Incomplete solution of a problem in Qualitative ODE.

I can't to complete my solution in the following problem: Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be $C^1$ and $q\in\mathbb{R}$ such that $f(q) = 0$ and $f'(q) >0$. Consider the Cauchy ...
0
votes
1answer
43 views

Solution of this ODE

How to solve the following ODE? $$i\partial_tv=t^q|v|^pv$$ where $i$ is the imaginary unit and $v$ is complex valued? I think that separation of variables is to be used.
0
votes
1answer
103 views

Find the general solution of $y'' + 9y = 0$

$y'' + 9y = 0$ and $y(0) = 0, \; y'(0) = 3$ Since this has real roots, I use the general solution $y_c = C_1 e^{r_1 t} + C_2 e^{r_2 t}$ I find the $y_c = \frac{1}{2}e^{3t}- \frac{1}{2}e^{-3t}$, but ...
0
votes
0answers
20 views

Method of undetermined coefficients for finite difference approximations

I'm reading over my text, and the first mention of deriving the coefficients states: "Suppose we want a one-sided approximation to $u'(x)$ based on $u(x), u(x-h)$, and $u(x-2h)$ of the form: ...
0
votes
0answers
27 views

How many terms to use in a Taylor series for local truncation error

So from my understanding for a finite difference approximation, you're supposed to expand the series "about" the point $x$, e.g., $$u(x+h) = u(x) + h \ u'(x) + ...
0
votes
1answer
59 views

How can I solve the ODE: $x^2y''+5xy'+4y=\dfrac{1}{x}$?

I'm given the ordinary differential equation $$x^2y''+5xy'+4y=\frac{1}{x},$$ which I'm trying to solve using variation of parameters. Now, if I know the auxiliary equation, I know the complementary ...
1
vote
0answers
21 views

Stability and Asymptotic Stability of Rational Matrix Solutions

If $X(t)$ is a fundamental matrix solution of $\dot{x}=A(t)x$ on $a<t<\infty$ and suppose the entries of $X(t)$ are rational functions of the variable t in the form $x_{ij}=p_{ij}(t)/q_{ij}(t)$. ...
0
votes
0answers
19 views

Uniform Stability on an Interval

I am trying to prove the following: Supposed $\phi (t)$ is a solution of $\dot{x}=f(t,x)$ defined on $(\alpha , \infty )$ and supposed $\alpha < \beta < \gamma$. They $\phi (t)$ is uniformly ...
5
votes
0answers
74 views

Uniqueness solutions of $dx/dt = f^2(x) + e^{-t}$.

Someone can help me in the following problem? Is a question of Zhang. Let $f(x)$ be continuous for $x \in \mathbb{R}$, show that $dx/dt = f^2(x) + e^{-t}$ has the property of uniqueness of ...
6
votes
2answers
64 views

Geometric series of an operator

In solving a first order linear differential equation $(1-D)y=x^2$ where $D\equiv \frac{d}{dx}$ the way I learnt was that we proceed as ...
0
votes
0answers
17 views

Variational methods

I have this BVP $$\begin{cases} u^{(4)}=f(t,u(t)),\, t\in[0,1]\\u(0)=u(1)=u''(0)=u''(1)=0\end{cases}$$ how to prove that the functional asociated to this BVP is $$J(u)=\frac12 ||u||^2-\int_0^1 ...
0
votes
0answers
21 views

Tangent solutions of $x' = f\left(\frac{x}{t}\right)$

Can someone help me in the following exercice? Lef $f:\mathbb{R}\longrightarrow \mathbb{R} $ be of $C^1$ class and $r\in\mathbb{R}$ such that $f(r) = r$. Show that If $f'(r) < 1$, ...
0
votes
0answers
21 views

derivation of higher order numerical methods for ODEs using Mathematica, Matlab, Maple

I want to know that is it possible to expand the multi-variable Taylor series of f(x+ah,y+ bhf(x+ch,y+dh)) in Mathematica. My purpose is to construct higher order methods which is very typical to ...
2
votes
0answers
77 views

Does the solution to this ODE have a closed form?

Consider the following two initial value problems: Problem 1: $\frac{dy}{dx}=\sqrt{\frac{1}{2\cos x}-\frac{y^2}{4}}, \ \ y(0)=-\sqrt{2}$ Problem 2: $\frac{dy}{dx}=\sqrt{\frac{1}{2\cos ...
0
votes
1answer
28 views

Equilibrium analysis of a Holling Type II differential Equation

I'm working on a problem which I am to determine what value of parameter $a$ will make a population become extinct. However, I am struggling in getting to the equilibrium points, which is necessary. ...
4
votes
2answers
115 views

Prove the Contraction Mapping Theorem.

Prove the Contraction Mapping Theorem. Let $(X,d)$ be a complete metric space and $g : X \rightarrow X$ be a map such that $\forall x,y \in X, d(g(x), g(y)) \le \lambda d(x,y)$ for some ...
1
vote
2answers
69 views

If a solution of an IVP is non-unique, then there are infinitely many solutions.

I am trying to prove than when an IVP has a non-unique solution then there exists infinitely many different solutions. I know that when the lipschitz conditions holds that there is at most one ...
4
votes
2answers
98 views

A first-order non-linear ordinary differential equation containing various squares

The Equation: Find all differentiable functions $f: I \rightarrow \mathbb{R}$ satisfies: $$\big(\,f(x)-x\,f'(x)\big)^2 = \big(\,f'(x)\big)^2 + 1 \; \; \; \; \; \text{for all}\,\,\, x \in I,$$ where ...
2
votes
0answers
44 views

Eigenfunctions.

I have the following ODE: $$y''-2xy'+2\alpha y=0$$ whose solution $y(x)$ may be recursively represented as: $$a_{n+2} = \frac{a_n(2n-2\alpha)}{(n+2)(n+1)}$$ I have found the eigenvalues to be ...
4
votes
2answers
70 views

How can we construct a differential equation from a system of differential equation?

Suppose we have a linear differential equation of order $n$. All of us know how to write it down as a system of linear differential equation as $X' = A_{n \times n} X_{n \times 1}$. My question is ...
2
votes
2answers
53 views

$xT' = 1$ in $D'(\mathbb{R})$

I need help solving the following problem: I want to show that all solutions of $$xT' = 1\ , T \in D'(\mathbb{R})$$ take the following form: $c_{1} + c_{2}1_{[0, \infty)} + ln|.|$ What I tried so far ...
0
votes
0answers
19 views

compute solution of second order bounday value problem [duplicate]

Let the second order boundary value problem $$-(e^{-2x}u')'-\ln(x^2+2)u= 2 e^ {-2x} - x \ln(x^2+2),\,\, x\in ]0,1[, u(0)=0,u(1)=1?$$ How we can find the exact solution for this problem? Thank's for ...
0
votes
0answers
100 views

Question on derivative

If $F_2(u)=\frac12 (Au,u)$ where $A$ is a continuous and self adjoint operator and $\eta$ the flow défini l'o.d.e $$ \begin{cases} \displaystyle\eta '(s)=- \frac{A\eta(s)}{||A\eta (s)||}\\ \eta(0)=u ...
0
votes
1answer
57 views

Find the solution for a boundary value problem

Please, how can we find the solution of this second order boundary value problem $$-(e^{-2x}u')'-\ln(x^2+2)u= 2 e^ {-2x} - x \ln(x^2+2),\,\, x\in ]0,1[, u(0)=0,u(1)=1?$$ Or more generally, What's the ...
5
votes
4answers
145 views

Solving a challenging differential equation

How would one go about finding a closed form analytic solution to the following differential equation? $$\frac{d^2y}{dx^2} +(x^4 +x^2+x+c)y(x) =0 $$ where $c\in\mathbb{R}$
7
votes
2answers
212 views

Solving the differential equation $f'(x)=af(x+b)$

How does one find all the differentiable functions $f\colon \mathbb{R} \to \mathbb{R}$ which satisfy the equation $$ f'(x)=af(x+b),\quad \text{for}\quad a,b \in \mathbb{R}? $$ I see that functions ...
0
votes
0answers
47 views

Continuation principle

In the following $g$ is a function related with some norms of solution of a certain differential eqution: $g$ be a nonnegative continous (if necessary, it is monotone increasing) function satisfying ...
0
votes
2answers
73 views

A first-order non-linear ordinary differential equation containing $(f')^2$

The Problem: Find all differentiable functions $f: \text{D} \rightarrow \mathbb{R}$ satisfies: $$f(x)\left [ 1-f'(x)^2 \right ] = 2xf'(x) \; \forall x \in \text{D}$$ , whereas $\text{D}$ is a/an ...
4
votes
1answer
146 views

Some Scaling Estimate for Heat Kernel

NOTE. I have rewritten the question to summarize my current progress on this question. The bounty is for completing what I have done so far, or by offering a more elegant solution probably based on ...