Differential Definition and 1000 Threads

hi, i am working on nonlinear differential equation- i know rules which decide the equation to be nonlinear - but i want an answer by which i can satisfy a lay man that why the word nonlinear is used. it is easy to explain nonlinearity in case of simple equation i.e when output is not...

A ##\frac{df}{dx}## notation is problematic. Obviously, the letter 'd' has very different meaning when applied to the function or to the argument. Additionally, a separate letter '##\partial##' is used to denote a partial differential (a very rare case in math when a notation used for a general...

Hello! First I tried modelling it like most mixing problems. $$ \frac{dA}{dt} = rate coming in - rate coming out $$ where dA is the volume and dt is the time rate coming in/out can be describe as; contrencation * flow rate. Now if we plug that all on $$ \frac{dA}{dt} = 35 * 0 -...

Problem statement: Second order linear differential equation in standard from Reasoning:

$s=c_1*\exp(-c_2*|(t)|)*r(t)$ But how can I solve $c_1+c_2$ ?

is the magnetic field B→. a state function and exact differential? I argued that it's a state function, what do you guys think

I had an equation. $$T=\frac{1}{2}m[\dot{x}^2+(r\dot{\theta})^2]$$ Then, they wrote that $$\mathrm dr=\hat r \mathrm dr + r \hat \theta \mathrm d \theta + \hat k \mathrm dz$$ I was thinking how they had derived it. The equation is looking like, they had differentiate "something". Is it just an...

The number of organisms in a population at time t is denoted by x. Treating x as a continuous variable, the differential equation satisfied by x and t is dx/dt= xe^-1/(k+e^-1), where k is a positive constant.. Given that x =10 when t=0 solve the differential equation, obtaining a relation...

Consider the second order linear ODE with parameters ##a, b##: $$ xy'' + (b-x)y' - ay = 0 $$ By considering the series solution ##y=\sum c_mx^m##, I have obtained two solutions of the following form: $$ \begin{aligned} y_1 &= M(x, a, b) \\ y_2 &= x^{1-b}M(x, a-b+1, 2-b) \\ \end{aligned} $$...

I attatched an example plot where I created the histogram for the differential distribution with respect to the energy of the d-quark produced in the scattering process. My conception is that the phase space generator can "decide" how much of the available energy it assigns to the respective...

My book claims that the diff. form of Gauss' law is $$\nabla\cdot\mathbf E=4\pi\rho$$ Can someone tell me why it isn't ##\nabla\cdot\mathbf E=\rho/\epsilon_0##?

Are these books using 'differential' correctly? Why not just write 'different'? 1. Handbook of the Biology of Aging edited by Edward J. Masoro, Steven N. Austad. p 480. 2. The Cambridge Companion to the Philosophy of Biology edited by David L. Hull, Michael Ruse. p 46. 3. p 78. 4. Dictionary...

Question about the differential in Calculus. Assume a function y = f(x) , differentiable everywhere. Now we have for some Δx Δy = f(x + Δx) - f(x) The differential of x, is defined as “dx”, can be any real number, and dx = Δx The differential of y, is defined by “dy” and dy = f’(x)...

Intersecting the graph of the surface z=f(x,y) with the yz -plane. This is the option I have chosen, but it's wrong. I don't understand why. x is fixed so I thought the coordinates: y and z are left. I thought this source may be helpful...

Hello. After a lot of researching, I am still not clear how the subject of differential equations is really any different from derivatives and integrals which are learned in the main part of calculus. For example: "Population growth of rabbits: N = the population of rabbits at any time t r=...

I assume that air have ##1 kg/m^3## density. Therefore, using Bernoulli equation, on upside and downside of my test object, there is a differential pressure ##\Delta P##: $$\Delta P=0.5*(v_2^2 - v_1^2)$$ From cases: (a) ##v_1 = 1 m/s## and ##v_2 = 2 m/s##, then ##\Delta P = 1,5 Pa## (b) ##v_1 =...

I was just browsing through the textbooks forum a few days ago when I came across a post on differential geometry books. Among the others these two books by the same author seem to be the most widely recommended: Elementary Differential Geometry (Barret O' Neill) Semi-Riemannian Geometry with...

Let ##\quad z=h(x, y)## and ##x=f(t) ; y=g(t)## Let the change in the function z be given by ##\Delta z=h(x+\Delta x, y+\Delta y)-h(x,y)## We can also write the change as ##\begin{aligned} \Delta z=h &(x+\Delta x, y)-\\ & h(x, y)-h(x+\Delta x, y) \\ &+h(x+\Delta x, y+\Delta y)...

Suppose you have a smooth parametrized path through spacetime ##x^\mu(s)##. If the path is always spacelike or always timelike (meaning that ##g_{\mu \nu} \dfrac{dx^\mu}{ds} \dfrac{dx^\nu}{ds}## always has the same sign, and is never zero), then you can define a smooth function of ##s##...

I am looking at this and i would like some clarity... at the step where "he let" ##μ_y##=0" Could we also use the approach, ##μ_x##=0"? so that we now have, ##μ_y##M=μ(##N_{x} -M_{y})##... and so on, is this also correct?

Let f be a 2 variables function. 1) ##f(x,y)=g(x)+h(y)\Rightarrow df=g'(x)dx+h'(y)dy\Rightarrow\int df=g(x)+k(y)+h(y)+l(x)=f(x,y),\textrm{ if } k=l=0## 2) ##f(x,y)=xy\Rightarrow df=ydx+xdy\Rightarrow\int df=2xy+k(y)+l(x)\neq f(x,y)## Why in the second case the function cannot be recovered ?

A rumour spreads through a university with a population 1000 students at a rate proportional to the product of those who have heard the rumour and those who have not.If 5 student leaders initiated the rumours and 10 students are aware of the rumour after one day:- i)How many students will be...

Hello ! I need to solve this diffrential equation. $$ y^{(4)} + 2y'' + y = 0 $$ First I wanted to find the homogenous solution,so I built the characteristic polynomial ( not sure if u say it so in english as well).I did that like this $$\lambda^4 +2\lambda^2+1 = 0 $$. The solutins should be...

I am currently pursuing a Bachelors in Physics. With my current work experience, that degree will eventually allow me to reach an engineering position in Non Destructive Testing. While I enjoy the career field I believe I could do more with my degree. I personally would like to work at LHC or...

How to solve this first order second degree differential equation ? ##\left(\frac {dy} {dx}\right)^2 + 2x^3 \frac {dy} {dx} - 4x^2y=0 ## Thanks.

Can anyone recommend a good on-line class for differential geometry? I'd like to start studying GR but want a good background in differential geometry before doing so. Many thanks.

I know there are more convenient differential elements that can be chosen to compute the moment of inertia of a disc(like rings). the mass of the differential element: $$dm = (M/\pi R^2) (dA) = (M/ \pi R^2) (2\sqrt{R^2 - y^2})(dy)$$ the moment of inertia of a rod through its COM is...

Any idea how to solve this equation: ## \ddot \sigma - p e^\sigma - q e^{2\sigma} =0 ## Or ## \frac{d^2 \sigma}{dt^2} - p e^\sigma - q e^{2\sigma} =0 ## Where p and q are constants.Thanks.

One thing that bothers me regarding the phase portraits, if I plot a phase portrait, then all my possible solutions (for different initial conditions) are included in the diagram? In other words, a phase portrait of a system of ODE's is its characteristic diagram?

When I was following the calculations of finding the potential energy of a spring standing on a table under gravity, I encountered the integral shown below, where ##d\xi## is the compression of a tiny segment of the spring and ##k'## is the effective spring constant of that segment. The integral...

Hi guys, I have just started studying DEs on my own, so pardonne moi in advance for the probably silly question :) Via Newton's second law of motion: $$x''=\frac{F}{m} \ [1]$$ Which is a second-order differential equation. But, from here, how do I get the good old equation of motion...

Hi guys, how are you doing? My maths teacher asked me to work on and deliver an engaging insight-oriented "lesson" to my class, about physical/engineering and real-world applications of differential equations, in order to better get the meaning of operating with such mathematical objects. Of...

There is an mass-spring oscillator made of a spring with stiffness k and a block of mass m. The block is affected by a friction given by the equation: $$F_f = -k_f N tanh(\frac{v}{v_c})$$ ##k_f## - friction coefficient N - normal force ##v_c## - velocity tolerance. At the time ##t=0s##...

Hello there, I need some advice here. I am currently studying intro physics together with calculus. I am currently on intro to oscillatory motion and waves (physics-wise) and parametric curves (calc/math-wise). I noticed that in the oscillatory motion section, I need differential equation...

Hi everyone, Imagine I have a system of linear differential equations, e.g. the Maxwell equations. Imagine my input variables are the conductivity $\sigma$. Is it correct from the mathematical point of view to say that the electric field solution, $E$, is a function of sigma in general...

(x cos(y) + x2 +y ) dx = - (x + y2 - (x2)/2 sin y ) dy I integrated both sides 1/2x2cos(y) + 1/3 x3+xy = -xy - 1/3y3+x2cos(y) Then I get x3 + 6xy + y3 = 0 Am I doing the calculations correctly? Do I need to solve it in another way?

In Miles Reid's book on commutative algebra, he says that, given a ring of functions on a space X, the space X can be recovered from the maximal or prime ideals of that ring. How does this work?

I am trying to self study Ordinary Differential Equations and am totally fed up of "cookbook style ODEs". I have recently finished Hubbard's Multivariable Calculus Book and Strang's Linear algebra book. I would like a rigorous and Comprehensive book on ODEs. I have shortlisted a few books below...

First time looking at differential forms. What is the difference of the forms over R^n and on manifolds? Does the exterior product and derivative have different properties? (Is it possible to exaplain this difference without using the tangent space?)

Summary:: solution of first order derivatives we had in the class a first order derivative equation: ##\frac{dR(t)}{dt}=-\sqrt{\frac{2GM(R)}{R}}## in which R dependent of time. and I don't understand why the solution to this equation is...

$\tiny{1.5.7.19}$ \nmh{157} Solve the initial value problem $y'+5y=0\quad y(0)=2$ $u(x)=exp(5)=e^{5t+c_1}$? so tried $\dfrac{1}{y}y'=-5$ $ln(y)=-5t+c_1$ apply initial values $ln(y)=-5t+ln(2)\implies ln\dfrac{y}{2}=-5t \implies \dfrac{y}{2}=e^{-5t} \implies y=2e^{-5t}$

hi guys i was trying to solve this differential equation ##\frac{d^{2}y}{dt^{2}}=-a-k*(\frac{dy}{dt})^{3}## in which it describe the motion of a vertical projectile in a cubic resisting medium , i know that this equation is separable in ##\dot{y}## but in order to solve for ##y## it becomes...

Using differential expressions for the generator, verify the commutator expression for ##[J_{\mu\nu},P_{\rho}]=i(\eta_{\mu\rho}P_{\nu}-\eta_{\nu\rho}P_{\mu})## in Poincare group Generator of translation: ##P_{\rho}=-i\partial_{\rho}## Generator of rotation...

Around the corner how a differential steering works 1975 ! Engineering Books Join our group : https://www.facebook.com/groups/2373375319542186/

I have tried to do it in standard way by integrating in PDE's but it turned out that ##\psi## is a function of y, so now I have no clue to start this. I know the range of ##\sqrt {g}y## from ##\frac{-\pi}{2}## to ##\frac{\pi}{2}##

By considering a vector triangle at any point on its circular path, at angle theta from the x -axis, We can obtain that: (rw)^2 + (kV)^2 - 2(rw)(kV)cos(90 + theta) = V^2 This can be rearranged to get: (r thetadot)^2 + (kV)^2 + 2 (r* thetadot)(kV)sin theta = V^2. I know that I must somehow...

kindly note that my question or rather my only interest on this equation is how we arrive at the equation, ##v(x)=ce^{15x} - \frac {3}{17} e^{-2x}## ...is there a mistake on the textbook here? in my working i am finding, ##v(x)=-1.5e^{13x} +ke^{15x}##

now my approach is different, i just want to check that my understanding on this is correct. see my working below;

Define that $$w(x,y,z)=zxe^y+xe^z+ye^z$$ So the constraint equation is ##x^2y+y^2x=1##. And its differential is ##dy=-\frac{2xy+y^2}{2xy+x^2}##. However, the solution plugs in ##z=0## when computing ##\frac{\partial w}{\partial x}## as shown in the screenshot below. While I understand that...

Hi, Could you please have a look on the attachment? Question 1: Why is this differential equation non-linear? Is it u=\overset{\cdot }{m} which makes it non-linear? I think one can consider x_{3} , k, and g to be constants. If it is really u=\overset{\cdot }{m} which makes it non-linear then...