Hi everyone,
The exam #3 review sheet is posted under “Classroom Resources/Exam Reviews”. Let me know if you have any questions or corrections.
Regards,
Prof. Reitz
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Differential Equations on Math.Stackexchange
- Nondimensionalization of given equationI have the following equation, which is a logistic system with a death term, $$\frac{dN}{dt}=rN(1-N/k)-p(N),$$ where $p(N)=\frac{BN^2}{A^2+N^2},$ for unknown constans $A,B$. I want to nondimensionalice the equation with a change of variables. I tried the typical change of variables for the nondimensionalization of the simple logistic equation, but i'm not being able to manage with […]
- Hypothesis on a parameter to ensure the unique solvability of a modified Laplace equationLet $\Omega\subset\mathbb R^3$ be a bounded Lipschitz domain, $n$ the normal vector on its boundary and $q\in L^{\infty}(\Omega)$. I want to find the minimal hypothesis on $q$ such that the following system has a unique solution in $H^1(\Omega)$ : $$ \begin{cases} (-\Delta + q) v= f&\mbox{ in }\Omega\\ \frac{\partial v}{\partial n}=0&\mbox{ on }\partial \Omega \end{cases} […]
- Let $f$ have compact support and $-u''+u=f$ a.e. on $(0, 1)$. Is the support of $u$ a compact subset of $(0, 1)$?I'm reading section 8.4 Some Examples of Boundary Value Problems in Brezis' Functional Analysis. Consider the problem $$ (14) \quad \left\{\begin{array}{l} -u^{\prime \prime}+u=f \quad \text { on } I=(0,1), \\ u(0)=u(1)=0, \end{array}\right. $$ where $f$ is a given function (for example in $C(\bar{I})$ or more generally in $L^2(I)$ ). The boundary condition $u(0)=u(1)=0$ is called […]
- What is the modeling of this system?enter image description here What is the modeling of this system? And can the modeling results be easily factored? condition: 1.The system consists of two springs, two objects and one damper. 2.The input is assumed to be a unit step function u(t), 3.Initial values are all set to 0, 4.No gravity. ($k_1$ on the bottom […]
- Prove that a particle moves in a planeLet's say $r \times v = const$ and $r \times a = 0$. We should prove that the particle moves in a plane. I thought to approach it from a differential equation perspective, where we know from the second equation that $a= r'' = kr$, which means that $r(t) = c_1e^{kt}+c_2e^{-kt}$. If $k$ would not […]
- Solve and integral equation with symmetric kernelI have the following integral equation with symmetric kernel $$x(t)=\sin(\pi t)+\pi \cos (\pi t) +\lambda \int_{0}^{1} k(t,s)x(s)\,ds $$ where $k(x,t)$ is a symmetric kernel given by $$k(t,s)= \begin{cases} (t+1)s, & 0 < t
- From equation $\frac{d^2\vec{r}}{dt^2} + 4\vec{r} = \vec{0}$, solve $\vec r$The motion of a particle is described by the position vector $\vec{r}$ at time $t$, following the equation $\frac{d^2\vec{r}}{dt^2} + 4\vec{r} = \vec{0}$. Determine the expression for $\vec{r}$ at any time $t$ under the given conditions $\vec{r} = \vec{a}$ and $\frac{dr}{dt} = 0$ at $t = 0$. Sol: Integrating the differential equation $(\frac{dr}{dt})^2+4\vec r^2=c$. Using […]
- Assistance with Similarity Solution for nonlinear PDE (porous linear equation)Consider $$\frac{\partial u}{\partial t}=\alpha\frac{\partial}{\partial x}\left(u\frac{\partial u}{\partial x}\right)$$ over the domain $$x,t>0$$ and the boundary condition, $$u(0,t)=t.$$ Using the similarity solution method, write $$u$$ in terms of a single variable $$f(\eta),$$ for some suitably defined variable $\eta.$ The question then asks us to obtain the ODE and boundary conditions governing $f(\eta)$, although we do not have […]
- Find a differential equation whose solution is a family of circles with centers at (h,k) and of fixed radius. Tenenbaum and Pollard Problem 1.4.20I know $r^2 = (x-h)^2 + (y-k)^2$ (A) Differentiate (A) $0 = 2(x-h) + 2(y-k)^2y'$ $0 = x - h + (y-k)^2y'$ (B) Differentiate B $0 = 1 + 2(y-k)y'^2 + (y-k)^2y''$ (C) At this point I can solve for h, solving for k I think I would have to expand the terms in (C). […]
- Determine if a bifurcation happens and what type it isFor $\dot{x}=f(\mu,x) = -\mu x + 2x^3 - x^5$, determine if (and what type) of bifurcation happens at $(\mu,x) = \{(-1,0),(0,0),(1,1)\}$. Attempt I found the stationary points as a function of the bifurcation parameter to be $x_0 = 0 $ and $x_{1,2,3,4} = \pm \sqrt{1\pm \sqrt{1-\mu}} $. In this case, because it's only one differential […]
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