Differential Definition and 1000 Threads

Homework Statement This problem is from V.I Arnold's book Mathematics of Classical Mechanics. Q) Show that every differential 1-form on line is differential of some function Homework Equations The differential of any function is $$df_{x}(\psi): TM_{x} \rightarrow R$$ The Attempt at a Solution...

Homework Statement Question is simple: what happens when ##DNL = -1 LSB## where DNL signifies differential nonlinearity and LSB stands for Least Significant Bit. It is also required to try and sketch such condition. Homework Equations Equation for differential nonlinearity: $$DNL(i) =...

Hi! I need your help for solving a couple of differential equation: dX/dt = a - b*X dY/dt = b*(c*exp(-E) - Y) - d*exp(-E)*Y X = X0 + f(Y, E) with X0, a, b, c and d are constants and f, a function of Z and E. Thank you in advance

Homework Statement This question relates to a very large project I have been assigned in a course on mathematical methods in structural engineering. I have to solve the following equation, in a specific way: (17) Now we have to assume the following solution: (18) It wants me insert...

Homework Statement We solved the differential equation (2.29), , for the velocity of an object falling through air, by inspection---a most respectable way of solving differential equations. Nevertheless, one would sometimes like a more systematic method, and here is one. Rewrite the equation...

I've a system of partial diff. eqs. in thermo-elasticity, I can solve it using normal mode analysis method but I need to solve it using laplace or Fourier

Homework Statement Suppose that a smooth differential ##n-1##-form ##\omega## on ##\mathbb{R}^n## is ##0## outside of a ball of radius ##R##. Show that $$ \int_{\mathbb{R}^n} d\omega = 0. $$ Homework Equations [/B] $$\oint_{\partial K} \omega = \int_K d\omega$$ The Attempt at a Solution...

Afternoon, anyone that would like to take a look at this Differential Equation problem it would be very helpful. I have tried separating the problem, but I am only working with one known term. Consider the logistic equation $$\dot{y}=y(1-y). $$ (a) Find the solution satisfying $y_1(0)=6$ and...

Homework Statement 2(y^2+1)dx+(4xy-3y^2)dy=0 Homework EquationsThe Attempt at a Solution here is my attempt to solve [/B]

Homework Statement Hello, I'm trying to go back to school and haven't done any math in awhile, as such, my skills are terribly out of practice. I am unable to arrive at the book's solution and suspect I am forgetting a simple algebraic trick and would like somebody to show me it and explain the...

Hi There is an example in my textbook worded as follows; A particle of mass 2kg moves along the positive x-axis under the action of a force directed towards the origin. At time t seconds, the displacement of P from O is x metres and P is moving away from O with a speed of v ms^-1. The force has...

Homework Statement how do we establish failure of uniqueness on this first order differential equation ## y(x)= x y'+(y')^2##Homework EquationsThe Attempt at a Solution [/B] general solutions are ## y= cx^2+c^2## where c = constant and ## y= -0.25x^2## ## -0.25x^2+cx+4c^2=0## ##x= -2c ⇒...

Moved from technical math section, so is missing the homework template. How to solve this equation please? I found charakteristic roots ia ##\pm \sqrt{-a^{-k^2}}##. Thank you Moderator note: Edited the LaTeX above to show the exponent correctly.

Let's say we have ##df=2xy^3dx + 3x^2y^2dy## - this is an exact differential. In integrating, to find f, can we write ## f = \int 2xy^3 \, dx + \int 3x^2y^2 \, dy = 2x^2y^3 + C ## Or am I getting it wrong?

I don't understand the following definition. If we let $u=\langle u,v \rangle$ , $p=\langle p,q\rangle,$ $x=\langle x,y \rangle$,then (x,y)=T(u,v) is given in vector notation by x=T(u). A coordinate transformation T(u) is differentiable at a point p , if there exists a matrix J(p) for which...

I know how to solve a differential equation using Eulers method but what if the equation has an integral part? i.e. a RLC electrical circuit. Vsource = iR + L di/dt + (1/C)int i dt can this be done? a link to how to solve this would be helpful.

So I've been Stuck On this Equation Trying to find the integrating factor (im not sure if it has one) appreciate the help

From my interpretation of this problem (image attached), the force applied to the point charge is equal and opposite to the repulsive Coulomb force that that point charge is experiencing due to the presence of the other point charge so that the point charge may be moved at a constant velocity. I...

Hello everyone, this is my first post here so please excuse me if its in the wrong area. I am looking for the tabular approach to solve a complex gear train based off the spur gear differential design. Attached is a photo of the differential. The requirements of the necessary differential...

Hello, I want to solve the following differential equation. $y'=\dfrac{x^3-y^3}{x-y}$. How to solve it?

Homework Statement I put this is the Calculus section because it relates to Calculus I and if I put it in Diff Eq section I think it would be assumed that I know the necessary terms, etc. My question is in regards to the use of the constant ##C## in differential equations. For reference, the...

I have currently been reading a book called 'Mathematical Methods In Physical Sciences'. Whilest I was looking at the differential section I came across a differential which I have never thought about before, which is of the form...

In the exercises on differential forms I often find expressions such as $$ \omega = 3xz\;dx - 7y^2z\;dy + 2x^2y\;dz $$ but this is only correct if we're in "flat" space, right? In general, a differential ##1##-form associates a covector with each point of ##M##. If we use some coordinates...

Im planning on taking a course on classical differential geometry next term. This is the outline: The differential geometry of curves and surfaces in three-dimensional Euclidean space. Mean curvature and Gaussian curvature. Geodesics. Gauss's Theorema Egregium. The textbook is "differential...

Salutations, I have been trying to approach a modelling case about organism propagation which reproducing with velocity $$\alpha$$ spreading randomly according these equations: $$\frac{du(x,t)}{dt}=k\frac{d^2u}{dx^2} +\alpha u(x,t)\\\ \\ u(x,0)=\delta(x)\\\ \lim\limits_{x \to \pm\infty}...

(I think I couldn't add the image) you can see my answer in link https://pasteboard.co/HPKZ6KD.jpg (Please first see my answer in the link) But in answer it is φ= Asin(kx) + Bcos(kx) I know that euler formula is eix = cosx +isinx But I can't get this answer can you help me?

Hi all, I am a bit new in this, am trying to learn DE, dynamical systems, & chaos. I am looking into some answers for the following questions: 1) Is it always possible to derive a difference equation for every differential equation, and if so how do we do that? 2) Consider Lorenz system...

Homework Statement Homework Equations The Attempt at a Solution Can someone check my answer please ?

Homework Statement Homework Equations The Attempt at a Solution [/B] Is my answer correct?

Homework Statement The AC response of an inductor can be modeled by the following differential equation: L \frac{di}{dt} + iR = V Find, using frequency response, the current of the system when the applied voltage V is: V = V_0 \sin(\omega t) Homework EquationsThe Attempt at a Solution In...

(a'[t]/a[t])^2 == K*(A + B*a[t]^-6)^1/2} is the equation to be solved for getting the solution of a(t) in terms of time(t). Any ideas on how to solve this problem? Use of Matlab or Mathematica is accepted.

Hello, $\vec{x'}=\small\begin{pmatrix}1&2\\3&2\end{pmatrix}\vec{x}+t\small\begin{pmatrix}2\\-4\end{pmatrix}$ Now i got the solution to this differential equation system as...

Homework Statement I need to solve the DE y’ = x^2y using the power series method Homework Equations y = sum(0->inf)Cnx^n y’ = sum(1->inf)nCnx^(n-1) The Attempt at a Solution I plug in the previous two equations into the DE. What is the general procedure for these problems after that...

Homework Statement Prove Sturm-Liouville differential operator is self adjoint when subjected to Dirichlet, Neumann, or mixed boundary conditions. Homework Equations l = -(d/dx)[p(x)(d/dx)] + q(x) The Attempt at a Solution I have no idea. If someone can give me a place to start that would...

Homework Statement The question is to solve the inexact equation by turning it into exact.the equation is ##( x + y + 4 ) d x + ( - x + y + 6 ) d y = 0## Where "x" and "y" are variable. 2. Homework Equations [/B] 1.(x+y+4)=m and (-x+y+6)=n 2.Integrating Factor =##\frac { 1 } { x ^ { 2 } + y...

Homework Statement Solve the following DE with the method of undetermined coefficients: y'' + 4y = 2cos(3x)cos(x) Homework Equations 2cos(3x)cos(x) = cos(4x) + cos(2x) The Attempt at a Solution Let's split the particular integral into two parts: yp1 and yp2. So yp1 is solution for RHS=cos(4x)...

I don't really understand how their inverses work. For example, in solving 2nd order linear non-homogeneous differential equations. The particular solution is found by ## y_{pi} = \frac{p(x)}{f(D)} ## And they continue by expanding using maclaurin series. How do you treat an operator as a...

Homework Statement [Taken from Razavi's Design of Analog CMOS Integrated Circuits 2nd edition]Homework EquationsThe Attempt at a Solution [/B] I'm not too sure what to do with this question. Here's what I think of the circuit This doesn't look like a differential amplifier to me since...

Homework Statement Solve the following differential equation such that ##x(0)=1##. ## \dfrac{dx}{dt} + 2tx = 3e^{-t^2}+t## Homework Equations Integrating factor: ##\mu(t) = exp\left(\int_0^t2t \right)## The Attempt at a Solution I used the integrating factor and then got the solution ##x(t) =...

Homework Statement The Attempt at a Solution I have deliberately made several obvious steps, because I keep ending up here. However I have no idea what to do from here. I thought about the fact, that differential equations have the solution ##x = x_{HOM} + x_{Inhom}##, but the ##x_{HOM}##...

4\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x} = 3u , u(x,0)=4e^{-x}-e^{-5x} let U =X(x)T(t) so 4X\frac{\partial T}{\partial t}+T\frac{\partial X}{\partial x} = 3XT 4\frac{\partial T}{T \partial t}+\frac{\partial X}{X \partial x} = 3 \left( 4\frac{\partial T}{T...

Hi. After arranging the dynamic contact between a elastic ball against a flat, I have reached the following differential equation for the motion during the contact: m·x’’+(k+c·x’)·x^n=0 with m,c,k>0 and for exponent n --> 1<n<2 Any functional form for this equation? I have solved it...

I'm learning Differential Geometry (DG) on my own (I need it for robotics). I realized that there are many approaches to DG and one is Cartan's, which is presented in Vargas's book. I think that book is highly opinionated, but I don't know if that's a good or bad thing. Does anyone of you know...

Hi, consider an "half-cone" represented in Euclidean space ##R^3## in cartesian coordinates ##(x,y,z)## by: $$(x,y,\sqrt {x^2+y^2})$$ It does exist an homeomorphism with ##R^2## through, for instance, the projection ##p## of the half-cone on the ##R^2## plane. You can use ##p^{-1}## to get a...

Homework Statement I have a coaxial cable with internal conductor of radius r1 and external conductor of radii r2 and r3. The material of the conductors has a conductivity ##\sigma_1##. Between the conductors there is a imperfect dielectric of conductivity ##\sigma_2##. Consider the...

We define the differential of a function f in $$p \in M$$, where M is a submanifold as follows In this case we have a smooth curve ans and interval I $$\alpha: I \rightarrow M;\\ \alpha(0)= p \wedge \alpha'(0)=v$$. How can I get that derivative at the end by using the definitions of the...

Hello, let $$M^n \subset \mathbb{R}^N$$ $$N^k \subset \mathbb{R}^K$$ be two submanifolds. We say a function $$f : M \rightarrow N$$ is differentiable if and only if for every map $$(U,\varphi)$$ of M the transformation $$f \circ \varphi^{-1}: \varphi(U) \subset \mathbb{R}^N \rightarrow...

Hello folks, I'm glad that I discovered this forum. :) You might save me. I'm hearing right now differential geometry and am having some problems with the subject. May you explain me the follwoing. We had the special case of the i-th projection. My lecturer now posited that the differential of...

Homework Statement acceleration of certain oscillating particle described by a = -x/9 determine the position of this particle when t = 3π/2 if when t=0 x=0 and v=v0 Homework Equations dv/dt=a The Attempt at a Solution frankly I am not sure how to start but i have two ways in my mind(even i...

Hi, I have to make an assignment on differential equations and Romeo and Juliet. r(t) is romeo's love for Juliet at time t, j(t) is Juliet's love for Romeo at time t So far, it is given: dr/dt=-j and dj/dt=r. It is also given that Romeo & Juliet's families are enemies, thus the initial...