1
vote
0answers
18 views

question about a proof: sequence of picard iteration converges uniformly

Given $u'(t)=t\cdot u(t)+t^3$ with $u(0)=0$ I want to show that $$u_{n+1}(t)=u(t_{0})+\int_{t_0}^{t}f(x,u_{n}(x))dx$$ converges uniformly on $[-b;b]$ and solves the differential equation. After ...
0
votes
1answer
30 views

Show the equality holds for any $x \in [0, \pi]$

We are considering a $2\pi$ periodic function defined on $x\in(-\pi,\pi)$ by $$f(x) = \pi - x, 0<x<\pi $$ and 0 otherwise. I already computed the full Fourier series is equal to: $$f(x) = ...
2
votes
1answer
25 views

Hyperbolic Systems ODE

Let $M_n$ the set of matrices of order $n \times n$ identified with $\mathbb{R^{n^2}}$ e $S=\{A \in M_n ; x'=Ax$ is hyperbolic$\}$. Show that $S$ is open and dense $M_n$.
0
votes
0answers
29 views

Show $u(x,t)$ is analytic in time

$$u_t + u_x + u u_x - u_{xxt} = 0$$ {know: $u$ can be differentiated $\infty$ times with respect to $t$. this fact may or may not be helpful in the proof} how would one approach such problem? i ...
1
vote
1answer
26 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 ...
0
votes
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
1
vote
1answer
38 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 $\mathbb{R}$) ...
1
vote
2answers
139 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 ...
3
votes
2answers
143 views

Real analysis question involving a linear ODE

Where do I start with this one? This question is really quite difficult..
0
votes
1answer
29 views

Showing that some differential equation has an infinite dimensional solution space?

I don't see how to proceed or even where to start to show this thing that I have found: The differential equation $$(\sin x)\frac{dy}{dx} - 2(\cos x)y = 0$$ has an infinite solution space of ...
20
votes
2answers
308 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
20 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 ...
1
vote
2answers
15 views

Lipschitz condition and continuity

I was wondering, if a function of one real variable is bounded on a compact interval and is $C^2$, is it necessarily true that the function is Lipschitz on that interval? Thanks
1
vote
2answers
76 views

Solutions for an ODE

I am looking for a solution of the ODE $x'(t)=x(t)+\frac{1}{1+e^{t}}$ which has finite limit when $t\rightarrow \infty$, I already find that the solutions are $e^t \ln(1+e^t)-te^t-1$ however these ...
0
votes
0answers
28 views

Uniqueness of solution to differential equation

First of all, I want to say that I need only a hint or guidance, rather than a direct solution. I was wondering, when the question says: given ~~~ differential equation, show that the expression of ...
1
vote
0answers
13 views

Flux and trajectories through vector field

I have here a very simple vector field $F(u, v) = 2\pi \binom{a}{b}$ where a and b are fix and $a \neq 0$. I have a parametrization of the surface of the Torus T with $\Phi(u, v) =$ $$ ...
1
vote
1answer
29 views

Approximation of the solution of an IVP

Consider the initial value problem $$\frac{dy}{dx} = x^2 + y^2, \\ y(0) = 0$$ on D = {|x| <= 1, |y| <= 1} Find the third approximation to the solution If someone could maybe walk me through ...
11
votes
2answers
165 views

If $f(x) + f'(x) + f''(x) \to A$ as $x \to \infty$ then show that $f(x) \to A$ as $x \to \infty$

This problem is an extension to the simpler problem which deals with $f(x) + f'(x) \to A$ as $x \to \infty$ (see problem 2 on my blog). If $f$ is twice continuously differentiable in some interval ...
0
votes
0answers
12 views

Initial value problem with some condition

Suppose $|f(t,x)|\le g(|x|)$ for some positive continuous function $g$ that satisfies $\int_{0}^\infty 1/g(r)dr=\infty$. Then, all solutions of the $x'=f(t,x)$ with $x(t_0)=x_0$ are defined for all ...
2
votes
2answers
64 views

Want to show that a solution of some ODE is bounded

Suppose that $u(t)$ satisfies the differential equation $$\dot{u}(t)=a(t)[u(t)-\sin(u(t))]+b(t),\;u(0)=u_0$$ for all $t\in\mathbb R$. In addition suppose that $a,b$ are continuous integrable on ...
0
votes
2answers
65 views

Eigenvalues of $d/dx$.

Consider $d/dx:C^\infty(\mathbf{R})\rightarrow C^\infty(\mathbf{R})$ (both as real vector spaces). I want to find its eigenvalues and corresponding eigenvectors. Every $\lambda\in\mathbf{R}$ is an ...
0
votes
1answer
22 views

Two initial conditions on the functions

Let $g \, : \, [0,1] \, \longrightarrow \, \mathbb{R}$ a function continuous on $[0,1]$. I would like to solve the following problem : $$ (S) \; \left\{ \begin{array}{l} f''=g \\ f(0)=a \\ f(1)=b \\ ...
1
vote
0answers
64 views

strong maximum principle for $u$ such that $u'' \geq c(x)u$

I want to prove that if $I$ is an interval in $R$, $c(x) \geq 0$ is continuous and $u \leq 0$ is $C^2$ then if for all $x$ $$u''(x) \geq c(x)u(x)$$ then the strong maximum principle holds for $u$, ...
0
votes
1answer
25 views

Derivative of a homogeneous function of degree n

How do we show that, if $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ is homogeneous of degree $d > 1$, and if $f$ is differentiable at $0$, then $f'(0)$ is the zero map from $\mathbb{R}^n$ to ...
3
votes
0answers
111 views

Proving Nonhomogeneous ODE is Bounded

I am trying to prove the following: Given a solution $x(t)$ of the IVP $\dot x=A(t)x+h(t)$, where $A(t), h(t)$ are continuous on $0<t<\infty$, prove that x(t) is bounded for $t\ge1$ if both ...
0
votes
2answers
76 views

A solution for a system of differential equations?

I want to check answer for specific ODE solvers, for instances, solving: $x_1' = 1/5\; x_1 + 4/5\; x_2$ $x_2' = 4/5\;x_1 + 1/5\; x_2$ $x_1(0) = 1$, $x_2(0) = 3$ I've just learnt how to solve these ...
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
0answers
19 views

Find examples where $\omega (\overrightarrow x)$…

Help figure examples where The set of all $\omega$ - points of $\phi_t (\overrightarrow x$) is called the $\omega$ - limit set of $\phi_t (x)$. 1) $\omega (\overrightarrow x) = \emptyset$ for all ...
1
vote
1answer
60 views

Separation of variables won't work

Find all solutions on $\mathbb{R}$ of the differential equation $y'=3|y|^{2/3}.$ Why wouldn't separation of variables method work for this differential equation? Why does the initial condition ...
0
votes
1answer
31 views

Find all solutions to a particular differential equation

Find all solutions on ${R}$ of the differential equation $ y' = 3|y|^ \frac{2}{3} $ I believe I need to use separation of variables, but it will only work if the initial condition is nonzero. ...
2
votes
2answers
57 views

Solving a differential equation $\displaystyle \frac{d \alpha}{dt}=w \times\alpha$

Let $\alpha$ be a regular curve in $\mathbb{R}^3$ such that $\displaystyle \frac{d \alpha}{dt}=w \times\alpha$ for $w$ a constant vector. How can we determine $\alpha$ ? $\displaystyle w ...
0
votes
2answers
54 views

If f plus the integral of f has a finite limit, show that f tends to zero

This is a problem from Pugh's Real Mathematical Analysis. Let $f$ be a continuous real-valued function on $[0, \infty)$ such that $$\lim_{x \to \infty} \left( f(x) + \int_0^x f(t) \, dt \right)$$ ...
1
vote
2answers
38 views

Given a solution flow to find periodic solutions

Given the system of differential equations $x' = 2x + y^3$ and $y' = -y$ i found the flow $$\phi_t(x,y) = ((x_0 + 1/5y_0^3)e^{2t} - 1/5 y_0^3e^{-3t}, y_0 e^{-t})$$. I am wondering are there any ...
0
votes
0answers
20 views

A set $A \subset \mathbb{R}^2$ containing more than one point that is invariant under the flow $\phi_t(x,y) = (xe^{-t},ye^{4t})$

A set $A \subset \mathbb{R}^2$ containing more than one point that is invariant under the flow $\phi_t(x,y) = (xe^{-t},ye^{4t})$ A set $A \subset \mathbb{R}^2$ containing more than one point that is ...
1
vote
1answer
32 views

A inital value problem $x' = f(x), x(0) = 1$ that has more than one solution?

A inital value problem $x' = f(x), x(0) = 1$ that has more than one solution? is this possible? if so, could you show me an example?
0
votes
1answer
56 views

Prove existence and uniqueness of differential/integral equation

This is a homework question, so I'm essentially asking for hints and not answers due to academic honesty concerns. The course is in real analysis (baby rudin) and it is essentially chapter 9 which ...
1
vote
2answers
124 views

Free fall with resistance: solution to the ODE

I'm having trouble solving this ODE: $$\ddot x = \mu \dot x^2 - g, \space \space x(0)=x_0$$ This is the ODE that determines the equation of motion of an object with air resistance. $\mu$ is a ...
0
votes
1answer
32 views

$\textbf c(t)$ is a flow line on $\textbf F = -\nabla V$, prove $V(\textbf c(t))$ is a decreasing function of $t$.

Let $\textbf c(t)$ be a flow line of a gradient field $\textbf F = -\nabla V$. Prove that $V(\textbf c(t))$ is a decreasing function of $t$. We have not learned Line Integrals, so I would assume this ...
0
votes
0answers
41 views

Find sufficient and necessary conditions such that the solution $u$ of this PDE is unique

Let us consider the following PDE probem: $$Δu(x,y)=\frac{\partial^2u(x,y)}{\partial x^2}+\frac{\partial^2u(x,y)}{\partial y^2}=0, (x,y)∈(0,1)\times\mathbb{R}$$ $$u\left(\frac{1}{2},y\right)=0, ...
2
votes
0answers
41 views

Exponential representation of picard iteration.

This is a homework question for a first course in real analysis (tiny Rudin) so I'd appreciate hints whilst straight out answers are discouraged due to academic honesty. I'm given recursively ...
2
votes
1answer
36 views

Finding all solutions to an initial value problem

This comes from a real analysis class, and I currently cannot assume any knowledge about integration. I want to find all solutions to the initial value problem $y' = y^{\frac{1}{2}}$, $y(0) = 0$. I ...
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 $x_{ij}=p_{ij}(t)/q_{ij}(t)$. ...
1
vote
0answers
18 views

Inverse Laplace Transform without tables

How can I show that the Inverse Laplace transform of $\dfrac{a}{s^2+a^2}=\sin at$ without the use of tables? Is there a formula or a direct way to do this?
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 ...
1
vote
0answers
170 views

Prove that $\vec x = \vec0$ is an asymptotically stable fixed point for this linear system.

Consider the linear vector field $\vec x' = A\vec x$, $\vec x \in R^2$ , where A is an $2$ x $2$ constant matrix. Suppose all eigenvalues of A have negative real parts. Prove that $\overrightarrow x = ...
5
votes
2answers
270 views

A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem: Let $f(x):\Omega\longrightarrow \mathbb R^n$ a $C^2$ function where $\Omega$ is an open subset of ...
1
vote
1answer
52 views

Prove that if $f:U \rightarrow \mathbb{R}$ is differentiable on $U$ then it is Lipschitz on $U$.

Let $U \subset \mathbb{R}$ be a closed and bounded set. Prove that if $f:U \rightarrow \mathbb{R}$ is continuously differentiable on $U$ then it is Lipschitz on $U$. Could anyone show how to prove ...
1
vote
1answer
36 views

Let $y'' + p(x)y' + q(x)y = 0$ , where $p(x)$ and $q(x)$ are continuous. Prove that the zeroes of $y$ are isolated.

Let $p$ and $q$ be continuous, and let $y$ be any solution of $y′′(x) + p(x)y′(x) + q(x)y(x) = 0$ that is not identically zero. Then zeroes of $y$ are isolated, in the precise sense that for any ...
2
votes
1answer
31 views

Discuss the existence and uniqueness of solutions of the equation $X' = X^{a}$ where $1>a > 0$ and $x(0) = 0.$

Discuss the existence and uniqueness of solutions of the equation $X' = X^{a}$ where $1>a > 0$ and $x(0) = 0.$ First, I notice that this equation is not differentiable at x = 0. Therefore, the ...
0
votes
0answers
47 views

Bounding Solutions of System of Inhomogeneous Equations using Gronwall Inequality

I am trying to use Gronwall's Inequality to show that for the system $x'=A(t)x+g(t)$, $x(1)=\left( \begin{array}{c} 1\\ 1\\ \end{array} \right)$, where $A(t)=\pmatrix{ \frac{1}{2t^4} & ...