Differential Definition and 1000 Threads

This is my attempt: \frac{dy}{dx} = \frac{y^2 - 1}{x^2 - 1} \\ \int \frac{dy}{y^2 - 1} = \int \frac{dx}{x^2 - 1} \\ \ln \left| \frac{y-1}{y+1} \right| + C_1 = \ln \left| \frac{x-1}{x+1} \right| + C_2 \\ \ln \left| \frac{y-1}{y+1} \right| = \ln \left| \frac{x-1}{x+1} \right| + C Since y(2) =...

Here, M = ##siny*cosy +xcos^{2}y ## and N = x ## M_y = (1/2)cos(2y) -xsin(2y)## and ##N_x = 1## Theorems: If R = ## \frac{1}{N} (M_y - N_x) = f(x), then I.F. = e^{ \int f(x) dx} ## If R = ## \frac{1}{M} (N_x - M_y) = g(y), then I.F. = e^{ \int g(x) dx} ## Neither is holding true. What should...

So I heard a k-form is an object (function of k vectors) integrated over a k-dimensional region to yield a number. Well what about integrals like pressure (0-form?)over a surface to yield a vector? Or the integral of gradient (1-form) over a volume to yield a vector? In particular I’m...

Here is a snip of the fundamental relation: This is from the book "Intro to Smooth Manifolds" and in this section it is simplified down to F as a map between just the real spaces R^n -> R^m (as shown above). I understand the meaning of this relation, as the following: The rightside is the...

I am throwing a bachelor party for my brother, who is currently getting his PhD in Math at columbia, and as you might expect, he is not very much of a party animal. I want to throw him a party he’ll enjoy, so I came up with scavenger hunt in the woods, where every step in the scavenger hunt is a...

Attempt at solution: Writing the chain rule for ## E(V,T) ##: ## dE = \frac{\partial E}{\partial T}dT + \frac{\partial E}{\partial V}dV ## Then, integrating the differential: ## \int{ dE } = \int{ \frac{\partial E}{\partial T}dT } + \int{ \frac{\partial E}{\partial V}dV } ## If I put the...

Hi, I was trying to see if the following differential equation could be solved using Laplace transform; its solution is y=x^4/16. You can see below that I'm not able to proceed because I don't know the Laplace pair of xy^(1/2). Is it possible to solve the above equation using Laplace...

ov!347 nmh{1000} Convert the differential equation $$y''+5y'+6y=e^x$$ into a system of first order (nonhomogeneous) differential equations and solve the system. the characteristic equation is $$\lambda^2+5\lambda+6=e^x$$ factor $$(\lambda+2)(\lambda+3)=e^x$$ ok not real sure what to do with...

For a function ##f: \mathbb{R}^n \to \mathbb{R}##, the following proposition holds: $$ df = \sum^n \frac{\partial f}{\partial x_i} dx_i $$ If I understand right, in the theory of manifold ##(df)_p## is interpreted as a cotangent vector, and ##(dx_i)_p## is the basis in the cotangent space at...

ln(y) +y = ln(x) + x +C y=?

Given: ##y_{2} - y_{1}= 0.2m## ##Q= 1.5\frac{m^{3}}{min}## ##Q= 0.025\frac{m^{3}}{s}## After conversion ##D_{1}= 0.2m## After conversion ##D_{2}= 0.1m## After conversion ##r_{1}= 0.1m## ##r_{2}= 0.05m## ##p_{1} - p_{2} = \frac{1}{2}P(v_{2}^{2}-v_{1}^{2}) + Pg(y_{2}-y_{1})## Calculating...

dM/dY = x+2y+1 dN/dx = 1 (My-Nx)/n = 1 Integrating Factor => e^∫1dx= e^x (xye^x+ye^x+ye^x)dx + (xe^x+2ye^x)dy = 0 dM/dY =xye^x+e^x+2ye^x dN/dx = xye^x+e^x+2ye^x Exact ∫dF/dy * dy = ∫ (xe^x+2ye^x)dy F = xy*e^x + y^2*e^x + c(x) dF/dx = xy*e^x + y*e^x + y^2 * e^x + c'(x)...

I have the following problem: Use the total differential to calculate approximately the largest error at determine the area of a triangle rectangle (right triangle) from the lengths of the cathetus if they measure 6 and 8 cm respectively, with a possible error of 0.1 cm for each measurement. I...

So in my previous math class I spotted on my book an exercise that I couldn't solve. We had to find the general solution for the differential equation. This was the exercise: 4y'' - 4y' + y = ex/2√(1-x2) Can anyone tell me how to solve this step by step?

Good evening, https://pdfs.semanticscholar.org/688b/e703a59a4a0c6fc96b4e42c38c321cd4d5b8.pdf Do you know :PROJECTIVE METHODS FOR STIFF DIFFERENTIAL EQUATIONS I have to make a program to solve a first-order differential equation according to this method but I do not arrive despite my efforts...

The integral equation is T(x_3,t_3|x_1,t_1)=\int \text{d}x_2T(x_3,t_3|x_2,t_2)T(x_2,t_2|x_1,t_1) where T(x_3,t_3|x_1,t_1) is the probability density of a Markov process taking the value x_3 at time t_3 given that it took the value of x_1 at time t_1. So far so good. To derive the differential...

There are two forms of the Maxwell equations, one is the differential form, the other is the integral form. Which one is more useful?

I started by substituting the following anzatz: $$ \psi = \psi_e + \psi_1 $$ When ## |\psi_1| \ll 1 ##. Substituting the above into the equation yields: $$ \frac {d\psi_1} {dt} = -(1 + i\alpha)\psi_1 + \frac i 2 \frac {\partial ^ 2 \psi _1 } {\partial x ^ 2 } + i (\bar \psi_1 \psi_1 ^2 + \bar...

WHAT HAPPENS IS That I need to model the example of A Protein G example, using a function f in Matlab, but when I execute the script, the graphics I get do not correspond to those of the example. The problem is that I can not understand what the model seeks to represent, besides that I do not...

Hi, for ease of reference this posting is segmented into : 1. Background 2. Focus 3. Question 1. Background: Regarding (one, linear, second-order, homogeneous, ordinary, differential) equation describing the force in a non-driven, damped oscillation: F = m.a = -k.x - b.v F =...

here I am trying to find ##\frac{d}{dt}dx## where ##x(t)## is the position vector Now ##\frac{d}{dt}(v_x(x,y,z,t)dt)=\frac{dv_x}{dt}dt=\frac{\partial v_x}{\partial t}dt+\frac{\partial v_x}{\partial x}dx+\frac{\partial v_x}{\partial y}dy+\frac{\partial v_x}{\partial z}dz## Now dividing by ##dx##...

hello, now I'm working on a numerical method called: Projective Methods for Stiff Differential Equations: Problems with Gaps in Their Eigenvalue Spectrum. but I can not understand despite the hours of work that I spent on it I turn to you for help, applying this method on this exmple : y '= y...

. Above is the figure of the problem. I am trying to solve x(t) and differentiate it to obtain v(t); however, I have difficulty solving the differential equation shown below. $$ v(t)=\int a(t)dt=\int \frac{B(\varepsilon-Blv)d}{Rm}dt \Rightarrow \frac{dx}{dt}=\frac{B\varepsilon...

I have attached the two pages in my notes and I have the following question. 1. Where have the n_t*l gone in 9.9? (According to 9.5 why do they disappear?) 2. Why J_s=sigma_tot J_i? The dimension of flux is per m^2 and sigma is per area too, the dimension is not right...

The rate at which glucose enters the bloodstream is ##r## units per minute so: ## \frac{dI}{dt} = r ## The rate at which it leaves the body is: ##\frac {dE}{dt} = -k Q(t) ## Then the rate at which the glucose in the body changes is: A) ## Q'(t) = \frac{dI}{dt} + \frac {dE}{dt} = r - k...

How do I start this? I plugged the differential equation at wolfram alpha and it semmed too complicated for such an exercise. I've also looked at a sample of an answer on cheeg where the initial approach is to rewrite the equation as ##\frac{d}{dt} (\frac{\dot\theta^2}{2}-cos(\theta)) = 0## How...

Hello, I have to solve this second order differential equation. It's like a string vibrating equation but with a constant c: $$\frac{{\partial^2 u}}{{\partial t^2}}=k\frac{{\partial^2 u}}{{\partial x^2}}+c$$ B.C $$u(0,t)=0$$ $$u(1,t)=2c_0$$ c_0 is also a constant I.C $$u(x,0)=c_0(1-\cos\pi...

So to answer my question I need to reference another problem, I hope i won't get flagged for this... it is only to make a point about the way I am trying to approach this current problem. the previous problem stated: y''+2y'-y=10 so first I am finding Y_(homogeneous) and going straight to the...

Here is my attempt at the solution: Y1(t)=kY2(t)→e^(3t)=ke^(-4t)→(e^(3t))/(e^(-4t))=k→e^(7t)=k So I have found a constant multiple of Y2(t), its the whole "interval" part that I don't get. The interval is (0,1), I guess I don't really know what they are trying to say...are they saying from 0...

Hey, someone I know told me that the differential dy/dx= 24x/(2x+3) is not possible to solve... Is this true? If not what's the differential for it. This is my first year of calc in high school so my apologies if I butchered some of the terminology.

Hey all, it's been awhile since done any calculus or DE's but was trying out some modelling (best price price per item for bulk value deals as a function of time and amount), in the last line i have f(n,t) implicitly. Any pointers or techniques for solving such things?

Hello I need to derive this equation from Grittfith's quantum book $$ \frac{d^2y}{dr^2} = r^2y$$ I know I can use the characteristic equation: $$m^2 = r^2 \rightarrow y = e^{r^2}$$ but the answer should be: $$y=Ae^{\frac{-r^2}{2}} + Be^{\frac{r^2}{2}}$$ I know from Euler's formula that...

Help please, I need to solve this differential equation x\frac{\partial^2 U}{\partial x^2}+y\frac{\partial^2 U}{\partial y^2}=aU in Matlab (where "a" is a constant parameter, it can be taken by any), I wanted to use the Partial Differential Equation Toolbox, but I ran into a problem, the...

When using the separation of variable for partial differential equations, we assume the solution takes the form u(x,t) = v(x)*g(t). What is the justification for this?

I'm currently an undergrad student who had to take a break from school for over a year and its been around 3 or 4 years since I took Calc I - Calc III and Linear Algebra. I'm debating on taking a introduction to differential equations course as an elective that starts in a couple weeks when I...

Homework Statement Find an equation that defines IMPLICITLY the parameterized family of solutions y(x) of the differential equation: 5xy dy/dx = x2 + y2 Homework Equations y=ux dy/dx = u+xdu/dx C as a constant of integration The Attempt at a Solution I saw a similar D.E. solved using the y=ux...

Homework Statement integrate: ##-v(du/dy) = κ(d^2(u)/dy^2) ## to obtain: ## (-v/κ)y = ln(du/dy) + c## and finally: ##u = d + w*e^(-vy/κ)## Homework Equations ##-v(du/dy) = κ(d^2(u)/dy^2) ## ## (-v/κ)y = ln(du/dy) + c## The Attempt at a Solution ## (-v/κ)dy = d(u) ## which gives: ##...

Homework Statement Homework Equations euler ##e^{ix} = cos(x) + i*sin(x)## ##e^{-ix} = cos(x) - i*sin(x)## The Attempt at a Solution I'm starting with differential equations and I'm trying to understand this solution including complex numbers: First we determine the zeros. I understand that...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am reading Chapter 13: Differential Forms ... ... and am currently focused on Section 13.1 Tensor Fields ... I need some help in order to fully understand some statements by Browder in Section 13.1 ... ...

Andrew Browder in his book: "Mathematical Analysis: An Introduction" ... ... defines a differential form in Section 13.1 which reads as follows: In the above text from Browder we read the following: " ... ... A differential form of degree ##r## (or briefly an ##r##-form) in ##U## is a map...

Homework Statement On a certain island, there is a population of snakes, foxes, hawks and mice. Their populations at time t are given by s(t),  f (t), h(t), and m(t) respectively. The populations grow at rates given by the differential equations s'=(8/3)s - f - (1/3)h - (1/6)m f'=(2/3)s + f -...

Dear Users, I would like to ask you how can I plot d_sigma/d_omega and d_sigma/d_theta for any collision (for instance, proton and proton) using pythia event generator. I would be greatful if you could tell me how make it. Any ideas would be appreciated. Kind regards.

Homework Statement Solve the following differential equation: y' = y / [ x + √(y^2 - xy)] 2. The attempt at a solution Using the standard method for solving homogeneous equations, setting u = y/x, I arrive at the following: ± dx/x = [1±√(u^2-u) ]/ [u√(u^2-u)] which in turn, I get the...

I’m trying to derive the infinitesimal volume element in spherical coordinates. Obviously there are several ways to do this. The way I was attempting it was to start with the cartesian volume element, dxdydz, and transform it using $$dxdydz = \left (\frac{\partial x}{\partial r}dr +...

Homework Statement (It should be noted that the actual problem has specific values associated with a, b, and c. However, at this point I'm trying to find a method to solve the problem rather than a specific solution). Homework Equations The Attempt at a Solution When I was trying to solve...

Homework Statement x(dy/dx) = 3y +x4cos(x), y(2pi)=0 Homework Equations N/A The Attempt at a Solution I've tried a couple different ways to make this separable, but you always carry over a 1/dx or 1/dy term and I can never fully separate this. I've also tried to do a Bernoulli differential...

I'm struggling to understand the importance of the differential cross-sectional area in Rutherford's scattering experiment, dσ/dθ. In one part of my course notes it is explained as 'the number of scatterings between θ and θ + dθ per unit flux, per unit range of angle'. However, dσ itself is...

I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Chapter 8, Section 2, Differential Forms ... I need some help in order to fully understand some statements of Shifrin at the start of Chapter 8, Section 2 on the dual space ... The relevant text from...

Homework Statement Hi, I'm trying to calculate the formula for the position vs. time of a rocket landing from an altitude of 100km. I'm neglecting a lot of forces for simplification but basically, I want to solve ##F_{net} = Drag - mg##. Homework Equations Drag Force: D = ## \frac {C_dAρv^2}...

I know this may sound as a stupid question but I would like to clarify this. An arbitrary function f can be expressed in the Fourier base of sines and cosines. My question is, Can this method be used to solve any differential equation? You plug into the unkown function the infinite series and...