Hi everyone,
Exam #2 will be returned in class tomorrow. If you are a member of this OpenLab site, you can find your grade under Dashboard/OpenLab Gradebook.
Regards,
Prof. Reitz
"...how it differs from the rocks"
Differential Equations on Math.Stackexchange
- What Exactly Is Symplecticity in the Context of Hamiltonian Systems and Numerical Schemes?I've encountered the term symplecticity in the study of Hamiltonian systems and their numerical discretizations, but I'm struggling to fully grasp its meaning and significance. Initially, I thought symplecticity was just a weaker form of the conservation of the Hamiltonian, but I've since learned that this isn't entirely accurate. From my current understanding, in the […]
- How to solve this equation with square roots:?I encountered this equation while working on a mathematical problem related to rational and square root functions. $$\sqrt{\frac{x+1}{x-1}} + \sqrt{\frac{x}{x+3}} = f(x)$$ For now, I attempted squaring both sides, but the solution seems complicated. Is there a systematic approach to simplify or solve this?
- Verifying Lipschitz Condition for a Function Involving Trigonometric and Rational TermsI have this ode: $$v''(r)+\frac{1}{2}(v'(r))^2+P(r)v'(r)=Q(r),$$ where $$P(r)=− \frac{k^3r + 12\pi \rho_0 kr \cos(kr) + 4\pi\rho_0(k^2r^2 − 3)\sin(kr)} {r[k^3r − 8\pi\rho_0[\sin(kr) − kr\cos(kr)]},$$ $$Q(r)= \frac{8\pi\rho_0kr(r^2 + 5) \cos(kr) + 8\pi\rho_0(k^2r^3 + k^2r^2 − r − 5) \sin(kr)} {k^3r^3 − 8\pi\rho_0r^2[\sin(kr) − kr \cos(kr)] }.$$ I wrote the equation as systems of ordinary differential equations: $$\begin{align}v'(r) &= […]
- generator of a semigroup properties $A$ and $A-\alpha I$at the moment I'm understanding a proof of a theorem about generating a C0-semigroup. Here's the theorem and the proof with associated definitions to understand it (see below). My question is what Hille-Yosida has shown that the operator $E_{1}^{-1}\tilde{A}_{1}$ generates a C0-semigroup. I don't understand why the author of the article directly deduces the thesis […]
- axisymmetric ODE $h=\frac{1}{r}\frac{d}{dr}\left(\frac{rh'}{\sqrt{1+(h')^2}}\right).$Consider the following equation where $h$ is a function of $y$: \begin{equation*} h=\frac{d}{dy}\left(\frac{h'}{\sqrt{1+(h')^2}}\right). \end{equation*} Multiplying both sides by $h'\equiv dh/dy$, and integrating the above equation, we get \begin{equation*} \frac{1}{2}h^2=-\frac{1}{\sqrt{1+(h')^2}}+c, \end{equation*} where $c$ is a constant of integration. Now consider the axisymmetric version of the same problem where $h$ is now a function of $r$, and […]
- Why is $X e^{\Lambda t}X^{-1}$ the general solution to system of $n$ linear first order ordinary differential equations?I know that a system of n first order linear ordinary differential equations: $$ \begin{align} \begin{bmatrix} \frac{dy_1}{dt} \\ \vdots \\ \frac{d^ny_n}{dt^n} \end{bmatrix} &= \begin{bmatrix} a _{11} & \cdots & a_{1n} \\ \vdots & & \vdots \\ a_{m1} & \cdots & a_{mn}\end{bmatrix} \begin{bmatrix} y_1 \\ \vdots \\ y_n\end{bmatrix} \\ \mathbf{Y}' &= A \mathbf{Y} \end{align} $$ is […]
- condition to fit the derivative inside an operator $\dfrac{d}{dt} Ex(t) = E\dfrac{d}{dt} x(t)$ (abstract ODE)I have an operator $E: X \longrightarrow Y$ where $X$ and $Y$ are Hilbert spaces and $E$ is a linear and bounded operator, but not necessarily inversible. I would like to know under what conditions can we fit the derivative inside the operator? that is to say when $\dfrac{d}{dt} Ex(t) = E\dfrac{d}{dt} x(t)$ where $x(t)\in […]
- Zero matrix for linearized systemI am given the following system, which I have to linearize such that I can describe the local phase portrait. $x' = x^2, y'=y^2$. The origin is the only equilibrium. If I compute the Jacobian of this system, I get $ \begin{pmatrix} 2x & 0 \\ 0 & 2y \end{pmatrix} $. Evaluated at $(0,0)$, I […]
- Finding the Steady-State Solution of the Burgers' EquationI am trying to determine the steady-state solution of the following Burgers' equation: \begin{equation*} u_t+u u_x=\nu u_{x x}, \quad u(x, 0)=\sin (\pi x) \end{equation*} subject to the boundary conditions $u(0, t)=u(1, t)=0$. Could someone provide guidance on how to find the steady-state solution for this equation? Any tips or references for solving such problems would […]
- Second order differential equation with lnI have a differential equation I need help solving $$f''(x)-(a^2+B\ln(x))f(x)=0$$ where a and B are constants. I know f(x) must be e to the power of something but no exponents works correctly. I got the equation from a Schrödinger equation after I plugged in the potential energy. Is there even an analytical solution? I'm not […]
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