## Examples of solutions of an ODE with this property November 29    1

Let $f:\mathbb R\times \mathbb R^n\to \mathbb R^n$ be a $C^1$ class function and suppose that $\varphi(t)$ defined in $\mathbb R$ is a solution of $x'=f(t,x)$, $x(t_0)=x_0$.I've been thinkin ...

## Variation of parameters, in one case we assume something in one we don't, but we end up with the same result November 29

You have a second order differential equation $y''+P(t)y'+Q(t)y=f(t)$. And you have two independent solutions to the homogeneous system, call them $y_{h_1}, y_{h_2}$. In order to find a particular solution to the nonhomogeneous system, you assume tha

## Solution of $\begin{cases} y(t)=z''\\ z(t)=y'' \end{cases}$ November 29    1

Solve $$\begin{cases} y(t)=z''\\ z(t)=y'' \end{cases}$$$y(0) = z(0) = 0$ $y(\pi/2) = z(\pi/2) = 1$ My attempt:$$\begin{cases} y(t)=z''\\ z(t)=y'' \end{cases}\Rightarrow y(t)=y^{(4)}$$The roots of corresponding characteristic equation are $1,-1,i,-i$

## How do I solve the DE $y''+6y'+8y = 2t+e^t$ November 29    1

The differential equation I am trying to solve is $$\dfrac{d^2y}{dt^2} + 6\dfrac{dy}{dt} + 8y = 2t +e^t$$ I know how to start off. I have done the $s^2 + 6s + 8 = 0$ to get $s = -4$ and $s = -2$ and have the $$y_p(t) = k_1e^{-4t} + k_2e^{-2t}$$Wh

## Uniform Continuity: i'm right November 29    1

Prove that $f : A\subset \mathbb{R} \to \mathbb{R}$ is uniformly continuous in $A$ $\iff$ for allthe sequences $(x_n), (y_n)\in A$ such that $$\lim_{n\to +\infty} ( x_n - y_n )= 0$$then $$\lim_{n\to +\infty} ( f(x_n) - f(y_n)) = 0.$$$$Proof$$Let's s

## Show that a linear form $\mathbb{R}^n \to\mathbb{R}$ is continuous November 29    1

$f(x)$=$n∑k=1$ $g$($x_k$) ou $x_k$ is the kth component of the vector x. $x_k=\langle e_k,x\rangle$.I have the option of showing this with sequences (which I dont know how, I never understood how to show that a function is continuous using sequences)

## Determining whether $f(x) = \frac{\sinx}{e^{x}-1}$ for $x \neq 0$, $f(x) = 1$ for $x = 0$ is continuous at $0$ November 29    3

$f: \mathbb R^m \to \mathbb R$ is defined as$$f(x) = \begin{cases}\dfrac{\sin||x||}{e^{||x||}-1} & \text{if x \ne 0} \\ 1 & \text{if x = 0.}\end{cases}$$Note that $x$ is a vector in $\mathbb R^m$.Is $f$ continuous at $0$? Well, $\|0\| = 0$. Ho

## Closed form of an integral November 29    1

Is there a closed form of $$\int\limits_0^1\frac{\arctan(\sqrt{x^2+2})}{(1+x^2)\cdot(\sqrt{x^2+2})}dx \quad \text{?}$$I just know that $\int\limits_0^1\frac{\arctan(\sqrt{x^2+2})}{(1+x^2)\cdot(\sqrt{x^2+2})}dx = 0.514042...$This is called the Ahmed i

## Closed form solution to $\{a_n\}_{n=1}^{\infty} = 1,2,2,3,3,3,…$ November 29    1

I had thought about this sequence (where each positive integer $n$ shows up $n$ times) the other day and think I have a closed form solution. First of all we know that the last time that $k$ shows up in the sequence is at $a_{\frac{k(k+1)}{2}}$. We w

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