WeBWorK: Assignments 7-Trench-EulersMethod and 7b-Trench-ImprovedEulersMethod will be due on Tuesday 3/12 at midnight.
OpenLab: none
"...how it differs from the rocks"
Differential Equations on Math.Stackexchange
- How to emphasis on value $c$ in the definition of unit step functionLet $c\geq 0$. The unit step function $U(x-c)$ is defined by $$ U(x-c)= \begin{cases}0,\quad 0\leq x < c, \\ 1,\quad x\geq c. \end{cases} $$ As I didn't find which one of the followings can be used, so make me light which one is better to say? "The unit step function $U(x-c)$ at $c$ is defined […]
- Implicit differential equations and degenerate metricsI am currently exploring systems of implicit differential equations of the first and second order and seeking resources to better understand them. My interest arises in the context of a manifold with a degenerate metric. In such a case, when one naively attempts to derive the equations for parallel transport or geodesics, a challenge arises: […]
- Approximate solution of ODE: Estimate error of the solution with polynom and the expression of the ODEConsider the initial value problem $$ \begin{cases} u'(t)=f(t,u(t))\\[10pt] u(0)=u_0 \in \mathbb{R} \end{cases} $$ Let's assume this initial value problem has a unique solution of the form $$ u:[0,T]\to \mathbb{R}. $$ with $ T>0 $. Instead of considering the expression of the solution $ u $ I chose annother way by creating a taylor polynom $ […]
- How should I show that an approximate periodic solution of the original equation is the following?Homework problem: Consider the differential equation (which is a modified van der Pol equation) $\ddot{x}+\epsilon(x^4-1)\dot{x}+x+\epsilon\alpha^2 x^3=0, x(0)=a>0, \dot{x}(0)=0, 0\leq\epsilon
- Wronskian for nonhomogeneous ODEFor a second-order linear homogeneous ODE, the Wronskian of two solutions is either identically zero (if the solutions are linearly dependent) or never zero (if the solutions are linearly independent). I am curious about what can be said for the Wronskian of two solutions of a second-order linear nonhomogeneous ODE of the form: $$ y'' […]
- Show that the equilibrium point of the system $(\dot{x}, \dot{y})=(y,q(t)x)$ is not asymptotically stableMy problem Let's consider the two-dimensional non-autonomous system \begin{align} \frac{dx}{dt}&=y, \\ \frac{dy}{dt}&=q(t)x, \end{align} where $q(t)$ is not a constant function and is a continuously differentiable function. Prove that the zero solution $($i.e.the equilibrium point $(0,0))$ is not asymptotically stable in the Lyapunov sense. My idea: I suspect that the zero solution is non-attractive (i.e., the […]
- How to apply Frobenius' Method to ODE with multiple simultaneous singularitiesI'm trying to analyze the following ODE and get something closed form. $$ \phi'' + \frac{1}{\left( \ 1+2b\cos (\theta) \ \right)\left( \ 1+b \cos (\theta) \ \right) \sin (\theta)} \phi' + \frac{\left( \ 1+2b\cos (\theta) \ \right) \left( \ 1+b \cos (\theta) \ \right)+2b^2\sin^2 (\theta)}{\left( \ 1+2b\cos (\theta) \ \right)\left( \ 1+b \cos (\theta) \ […]
- How can we extend the derivative (with respect to IC) of $\ddot x = \dot x^2 +x^3$ from small $t$ to large $t$?Find the derivative with respect to $B$ for $B = 0$ of the solution of the equation $$\ddot x = \dot x^2 +x^3$$ with initial conditions $$x(0) = 0 \\ \dot x(0) = B.$$ -- V. Arnold A complete solution is here and here, though I am not familiar with their techniques and cannot understand […]
- Strictly Increasing Function With Periodic First Derivative [closed]I was tasked with the following question: Construct the functions F such that F has domain R F is differentiable F is strictly increasing. F′ is periodic with period 2π, and F′ is not constant. This is what I've been able to come up with so far: Let $a \in \mathbb{R}$ and assume $a \neq […]
- A priori bound of approximation solution in proof of Leray theoremI meet a problem when reading Chemin's Mathematical Geophysics. In page 45, to prove Leray theorem (about global weak solution of Navier-Stokes equations) he constructs an ODE: $$ (NS_{\nu,k})\begin{cases} \dot{u}_k(t)=\nu\boldsymbol{P}_k\Delta u_k(t)+F_k(u_k(t))+f_k(t),\\ u_k(0)=\boldsymbol{P}_ku_0. \end{cases} $$ After showing the existence of maximal solution $u_k$ of $(NS_{\nu,k})$ in $C^{\infty}([0,T_k);\mathcal{H}_k)$, he wants to prove $T_k=+\infty$, thus he gives an […]
© 2024 2019 Spring – MAT 2680 – Reitz
Theme by Anders Noren — Up ↑
Leave a Reply