Differential Definition and 1000 Threads

Hi, I am learning ODE and I have some problems that confuse me. In the textbook I am reading, it explains that if we have a separable ODE: ##x'=h(t)g(x(t))## then ##x=k## is the only constant solution iff ##x## is a root of ##g##. Moreover, it says "all other non-constant solutions are separated...

I learned that differential of e^x is same but what's so special about it? What makes is so special as it seems like a normal function to me other than the fact that e= sum of series of reciprocal of factorial numbers. What i want to ask is if e^x differential is e^x then do this rule apply to...

Hi PF! Suppose we have a differential area element ##dA##. This can be expressed as ##dx \, dy##. However, a change in area ##dA## seems different. Take positions ##x## and ##y## and displace them by ##dx## and ##dy## respectively. Then the change in area ##dA = (x+dx)(y+dy)-xy = xdy+ydx##...

The differential form of a function is \partial{f(x^1,...,x^n)}=\frac{\partial{f(x^1,...,x^n)}}{\partial{x^1}}dx^1+...+\frac{\partial{f(x^1,...,x^n)}}{\partial{x^n}}dx^nIs there (especially in General Relativity) differential of higher orders, like \partial^2{f(x^1,...,x^n)}? If so, how is it...

Hi folks, I was wondering if there are books that explain how to solve differential equations using infinite series. I know it is possible to do it since Poincaré used that method. Do you know which ones are the best? I find books on infinite series but they talk just about series...

<<Moderator's note: this is a spin-off of https://www.physicsforums.com/threads/how-long-to-learn-physics.891250/>> To Zapper. Some differential equations have no analytical solutions but some do. I remember a case where a co worker was using a computer to find the solution to x dot = 1 /...

The electromagnetic action can be written in the language of differential forms as ##\displaystyle{S=-\frac{1}{4}\int F\wedge \star F.}## The electromagnetic action can also be written in the language of vector calculus as $$S = \int \frac{1}{2}(E^{2}+B^{2})$$ How can you show the...

Let ##0##-form ##f =## function ##f## ##1##-form ##\alpha^{1} =## covariant expression for a vector ##\bf{A}## Then consider the following dictionary of symbolic identifications of expressions expressed in the language of differential forms on a manifold and expressions expressed in the...

Consider a curve ##C:{\bf{x}}={\bf{F}}(t)##, for ##a\leq t \leq b##, in ##\mathbb{R}^{3}## (with ##x## any coordinates). oriented so that ##\displaystyle{\frac{d}{dt}}## defines the positive orientation in ##U=\mathbb{R}^{1}##. If ##\alpha^{1}=a_{1}dx^{1}+a_{2}dx^{2}+a_{3}dx^{3}## is a...

I am trying to solve this equation: d/dx[dF(x)/dx] = [c(c+1)/x^2)F(x), where c is a constant. Do I still use the characteristic equation to solve this? EDIT: Is it solvable using Dawson's integral rule?

Hello, I have been working on a little movement system in a program called Game Maker: studio. The code works fine on the programming perspective, but something I did not expect happened: When I ran the code by adding to the speed while pressing a key, and every step passively subtracting from...

Hi folks, I am reading Poisson's Teatrise on Mechanics. In the introduction he talks about the infinitesimals. Let's say A is a first order infinitely small quantity, a differential of the first order, if the ratio of A to B is infinitely small too it means B is an infinitesimal of the second...

Homework Statement I'm not sure how to approach this. The question involves linear operators and a non-homogenous differential equation. Here is the question: https://s15.postimg.org/cdmw80157/Capture.png Homework Equations They are given in the question The Attempt at a Solution I really...

Why does the definition of a differential form requires a totally antisymmetric tensor?

Homework Statement Solve for the solution of the differential equation and use the method of variation of parameters. x`` - x = (e^t) + t Homework Equations [/B] W= (y2`y1)-(y2y1`) v1 = integral of ( g(t) (y1) ) / W v2 = integral of ( g(t) (y2) ) / W The Attempt at a Solution [/B] yc= c1...

Homework Statement dy/dx= 200-2y. y(0)=75 Homework EquationsThe Attempt at a Solution Do you move dx over and integrate. Do you just integrate it 200y-y^2+c

Hi. Is the Maxwell equation $$\nabla\cdot\vec{E}=\frac{\rho}{\varepsilon_0}$$ even true in the stronger form $$\frac{\partial E_i}{\partial x_i}=\frac{\rho}{3\cdot\varepsilon_0}\enspace ?$$ I guess not, since I haven't found a source suggesting this. But shouldn't the isotropic electric field...

Homework Statement ##dz=Mdx+Ndy## is an exact differential if ##(\frac{\partial M}{\partial y})_x=(\frac{\partial N}{\partial x})_y##. By invoking the condition for an exact differential, demonstrate that the reversible heat ##Q_R## is not a thermodynamic property. Homework Equations...

I just started learning Legendre Differential Equation. From what I learn the solutions to it is the Legendre polynomial. For the legendre DE, what is the l in it? Is it like a variable like y and x, just a different variable instead? Legendre Differential Equation: $$(1-x^2) \frac{d^2y}{dx^2}...

Dear Physics Forum personnel, Is it possible to learn differential geometry simultaneously while learning the relativity and gravitation? I have been reading Weinberg's book (currently in Chapter 02), but I believe that modern research in relativity is heavily based on the differential...

Hi all, I have derived a differential equation, which I don't know how to solve. I can do some numerical simulations, but would really be interested in, at least, knowing if an analytical solution exists, so would appreciate any help with it: (I have removed argument from y)...

Homework Statement A tank contains 60 kg of salt and 2000 L of water. A solution of a concentration 0.015 kg of salt per liter enters a tank at the rate 6 L/min. The solution is mixed and drains from the tank at the same rate. Find the amount of salt in kg at t = 3 hours Find the...

Homework Statement xy(dx)=(y2+x)dyHomework Equations integrating factor : u(x)=e∫p(x)dx standard form of linear DE: dy/dx + P(x)y=Q(x) standard form of bernoulli's differential equation: dy/dx + P(x)y=Q(x)yn change of variables v=y1-n The Attempt at a Solution xy(dy)=(y2+x)dx xy(dy/dx)=y2 +x...

The polynomial equation and it's private solution: $$(1)~~ay''+by'+cy=f(x)=kx^n,~~y=A_0x^n+A_1x^{n-1}+...+A$$ If i, for example, take ##f(x)=kx^3## i get, after substituting into (1), an expression like ##Ax^3+Bx^2+Cx+D## , but that doesn't equal ##kx^3##

d^2x/dt=0.01-0.01dx/dt =>x(t)=-100c1(e^-0.01t)+c2+t How do we find c1 and c2. Are they numbers or functions? d^2x/dt^2 instead of d^2x/dt gives the same solution, which means different c1 and c2

Homework Statement Solve the differential equation: (ex+1)cosy dy + ex(siny +1)dx=0 y(0)=3 Homework Equations none The Attempt at a Solution (ex+1)cosy dy + ex(siny +1)dx=0 (ex+1)cosy dy =- ex(siny +1)dx cosy/(siny+1)dy=-ex/(ex+1)dx ∫cosy/(siny+1)dy=-∫ex/(ex+1)dx using u sub on both the...

Homework Statement Solve each of the following differential equations: 4xydx + (x2 +1)dy=0Homework Equations None The Attempt at a Solution 4xydx + (x2 +1)dy=0 (x2 +1)dy=-4xydx dy/y=-(4xdx)/(x2 +1) ∫dy/y=∫-(4xdx)/(x2 +1) ln|y|=-2ln|x2+1| +C used u-sub on last step fo u=x2 +1

Homework Statement Find the velocity of v as a function of displacement x for a particle of mass m which starts from rest at x=0 and subject to the following force: F=F_0+kv You could say mv = F0*t + kx, but the answer in the back of the book is an equation that is only in terms of x and v...

Homework Statement I Have a differential equation y'' -xy'-y=0 and I must solve it by means of a power series and find the general term. I actually solved the most of it but I have problem to decide it in term of a ∑ notation! Homework Equations y'' -xy'-y=0 The Attempt at a Solution I know...

Hello,I have a department elective course called "Differential equations theory" but I have no idea if it is going to be useful for me as a physicist (I'm interested in the theory/ minor math). The description of the course is as follows : The fundamental theorem of existence and autism, linear...

1. Homework Statement Posted Homework Equations Posted The Attempt at a Solution Posted I just need the work to be checked.

Hello, I have a maybe unusual question. In a paper, I recently found the equation $$\mathcal{L}_v(v_i dx^i) = (v^j \partial_j v_i + v_j \partial_i v^j) dx^i$$ Where v denotes velocity, x spatial coordinates and \mathcal{L}_v the Lie derivative with respect to v. Now I'm an undergraduate who...

Homework Statement Write down the component form of the differential equations of motion of a projectile if the air resistance is proportional to the square of the speed. Are the equations seperated? Show that the x component of the velocity is given by \dot{x}=\dot{x}_0e^{^-\gamma s} where s...

Hello, I am currently taking ODE's and the class has an optional lab to accompany it. So far in the lab we've been doing some pretty basic stuff. But we've finally moved on to entering in differential equations, and I'm confused. 1. Homework Statement dydx+2x=2y How do I enter this equation...

Hello! Im trying to solve this second order differential equation: \begin{equation*} -\dfrac{d^2y}{dx^2}+\dfrac{3}{x}\dfrac{dy}{dx}+(x^2+gx^4+2)y=0 \end{equation*} Any idea? Maybe it could be converted to a Bessel-like equation (?) with an appropriate change of variables. The equation...

Homework Statement Homework Equations Leibniz notation: dy/dx = f(x) g(y) integral 1/g(y) dy = integral f(x) dx The Attempt at a Solution integral 1/y dy = integral sqrt (abs x) dx ln (y) = ? because sqrt (abs x) is not integrable at x =0 Then my thought is that y=0 is not unique

Homework Statement Solve the differential equation ##\displaystyle Cv^2 - mg = m\frac{d^2 y}{dt^2}## Homework EquationsThe Attempt at a Solution The problem is nonlinear, so we need to use unconventional methods. Specifically, if we can express the derivative of y with respect to v, then we...

Homework Statement Find a solution \bf{\phi} of the system $$y'_1(t)=y_1(t)+y_2(t)+f(t)$$ $$y'_2(t)=y_1(t)+y_2(t)$$ where f(t) is a continuous function $$\bf{\phi} (0)=(0,0)$$ Homework Equations A hint was given to define ##v(t)=y_1(t)+y_2(t)## The Attempt at a Solution Using the suggested...

I would like to begin my first exploration of the arts of differential geometry/topology with the first volume of M. Spivak's five-volume set in the different geometry. Is a thorough understanding of vector calculus must before reading his book? I read neither of his Calculus nor Calculus on...

Problem:Find the differential equation satisfied (i) by the equation of the family of tangents to y=x^2 and (ii) by the equation of the family of normals to y=x^2.

Homework Statement Find the values of m so that ##y = x^m## is a solution of ##x^2\frac{d^2y}{dx^2} - 3x\frac{dy}{dx} -12y = 0## Homework Equations ##y = x^m## ##y'=mx^{m-1}## ##y''=(m^2-m)x^{m-2}## The Attempt at a Solution So after plugging and chugging we get $$(m+2)(m-6)x^m = 0 $$...

Homework Statement [/B] Its been a pretty long time since I've taken differential equations and I'm expected to know the solutions to the kinds of DEs below for my fluid mechanics class. In class my professor worked a 2nd order DE: dy2/dx2 = -k2*y and told us the way to think about it was to...

This is an experiment I have wanted to do for a few years now but don't have the necessary equipment. GR tells us if you have identical objects with the same weight exactly when they are at the same temperature, then when one object is heated, it will weigh more. This is because the...

I have the following linear differential equations: ##A\dot{x} + By = 0## ##C\dot{y} + Dx = 0## Where x and y are functions of t, and A through D are constants. I can solve this fairly easily by differentiating the first equation, rearranging, and removing one of the variables, which gives me a...

Homework Statement The equation of motion of a particle is given by the differential equation ##\frac{d^2x}{dt^2} = -kx##, where ##x## is the displacement of the particle from the origin at time ##t##, and ##k## is a positive constant. 1. Show that ##x = A\cos{(kt)}+B\sin{(kt)}##, where ##A##...

Homework Statement [/B] (dy/dx)^2 = (1-y^2) / (1-x^2)Homework Equations Separating the variables I arrive with: dy/sqrt(1-y^2) = dx/sqrt(1-x^2) By integration on both sides by trigonometric substitution and putting it in a general solution: Arcsin y - Arcsin x = C The Attempt at a Solution If...

I have been stuck in this problem for two days now. I am starting DE this semester and I want to move ahead. so for this problem I attempted to use x=yv, then v=x/y. so I move dy/dx to the other side of the equation and divide by 2xy both sides, which leads to (3x^2-2y^2)/(2xy)=dx/dy. then...

Hi I understood above differential ## typo. RHS= 2*f ' (x,2z-x) but, what is answer of below equation? is this right? f ' ( g(z),2z-x) * g' (z) + f ' ( g(z),2z-x) *2

Homework Statement Let be E a normded vectoriel space, ##dim(E) = m \in \mathbb{N}^{*}##, I have to show that ##\exists \rho_{1}, \rho_{2} > 0 | \forall u \in L(E), ||u^{m} - Id| \leq \rho_{1} \Rightarrow |u - Id| \leq \rho_{2}##. Homework Equations Nothing. The Attempt at a Solution [/B] I...

Hello folks, Let me explain this. Much effort, time, methods and books is devoted to the science of finding solutions of differential equations. But I cannot find anywhere the reverse problem. Basically I am thinking about the work that Maxwell did finding the differential equations that...