WeBWorK: Assignments 14-InverseLaplaceTransforms, and 16-LaplaceIVP are due Tuesday, 5/24, at midnight.
OpenLab: OpenLab #3, Advice for the Future, is due on Tuesday, 5/21 (the day of the final exam).
"...how it differs from the rocks"
Differential Equations on Math.Stackexchange
- Improve differential equation to model room temperature given input energy and outside temperatureI am trying to develop a model that will model my dorm room temperature given outside temperature and supplied energy of two electric heaters. Eventually I want to automate my room heating. I have recorded the following data over the past couple of weeks: Indoor temperature Outside temperature Socket power of my two electric heaters […]
- Difficulty in Solving an ODE Limit Problem: Using the Tubular Flow TheoremI am studying ordinary differential equations (ODE) in a brazillian book and came across the following problem: Suposse f is $C^1$ and let $ x(t) $ be a solution to the differential equation $ x'(t) = f(x(t)) $ defined on the entire real line such that $\lim_{t \to +\infty} x(t) = z_0,$ where $z_0$is a […]
- Comparison theorems for odeSuppose that the functions $f$ and $g$ are continuous in the $\mathbb{R}^2$, and denote by $y_0(x), z_0(x)$ any solution of the initial value problems $$ y'(x) =f(x,y),y(x_0) = y_0 \\ z'(x) = g(x,y),z(x_0)=y_0 $$ respectively. if $f(x,y) \leq g(x,y)$ and $f,g$ are locally Lipshitz, how to prove in interval $[x_0,b)$,we have $y(x) \leq z(x)$?
- Solutions to $\ddot{\gamma} \times\gamma =0$ where $\|\gamma(t)\|=1$I was brushing up on geodesics and I need to find the solutions to $\ddot{\gamma}(t) \times \gamma(t) =0$ where $\|\gamma(t)\|=1$ and $\gamma: I\rightarrow \mathbb{S}^2$ a $C^1$ curve. My attempt is to say that the components of the vector $\gamma(t)$ denoted respectively $\gamma_i$ in cartesian coordinates (so by imbedding $\gamma$ in $\mathbb{R}^3$) satisfy for all $t\in […]
- How to Prove $N \subseteq \bigcup_P (X \times Y) \setminus P^*$Let $N$ be a compact, forward-invariant attracting neighborhood in a dynamical system on $X \times Y$, with an attractor $A = \mathrm{Inv}(N)$. Suppose $P$ ranges over the largest attractors in $N$, and $P^*$ is the repeller associated with $P$. How can one rigorously prove the decomposition: $$ N \subseteq \bigcup_{P} \left( (X \times Y) \setminus […]
- Traffic Flow Modelling From one Lane Into Two LanesI need to derive a model that demonstrates the flow from one lane splitting into two lanes. I am a bit stumped on how to start. I imagine it would be similar to the way light traffic in front of heavy traffic is modeled. I am unsure how or where to start with this, and […]
- Proving existence of a periodic orbitI'd like to show that there is a periodic orbit for the system $$ x' =20-x-\frac{4xy}{1+x^{2}}, y'=3x\left(1-\frac{y}{1+x^{2}}\right)$$ in the upper right quadrant. To do this, I would like to use the Poincare-Bendixson theorem. The only fixed point occurs at $(4,17)$ and it is repelling. I was able to find an outer rectangle which is positively […]
- How should I show that $F(x)\to\infty$ as $x\to\infty$ in this problem? (Please check my work.)Show that the equation $\ddot{x}+(x^2-1)x^2\dot{x}+x^3=0$ has a unique periodic solution. Here's my work: Proof: Consider the differential equation $\ddot{x}+(x^2-1)x^2\dot{x}+x^3=0$. By theorem, the equation $\ddot{x}+f(x)\dot{x}+g(x)=0$ has a unique periodic solution if $f$ and $g$ are continuous, and (i) $F(x)$ is an odd function; (ii) $F(x)$ is zero only at $x=0, x=a, x=-a$, for some $a>0$; (iii) […]
- Is this IVP ill-posed?I'm being asked to (i) use Euler's method in this question to approximate the solutions of an IVP, (ii) as well as to give its theoretical error bound. The IVP is defined as follows: $y'=\frac{1+t}{1+y}$ where $1\le t \le 2$ given $y(1)=2$ and $h=0.5$ For the second part of the question (ii), I'm also given […]
- Differential inequality whose constraint condition is at infinity [closed]$\phi(t), H(t)$ is defined on $[0, \infty)$, and $H(t)>\delta>0, \phi(t)>0$. Besides, $\phi(t)$ satisfy $$ \left\{ \begin{align} &(\ln \phi(t))' \le H(t)(1-\phi(t)) ~~~~~~~~~~~~(1)\\ &\lim_{t\rightarrow +\infty} \phi(t)=1 \end{align} \right. $$ Then, how to show $\phi(t)\le 1$ for all $t\ge 0$ ? The problem is from this problem. Under Yimin's help, I can deal the (1) with $\phi(1) \le […]
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