Differential Definition and 1000 Threads

Homework Statement Solve the following coupled differential equations by finding the eigenvectors and eigenvalues of the matrix and using it to calculate the matrix exponent: $$\frac{df}{dz}=i\delta f(z)+i\kappa b(z)$$ $$\frac{db}{dz}=-i\delta b(z)-i\kappa f(z)$$ In matrix form...

Homework Statement If d^2/dx^2 + ln(x)y = 0[/B]Homework Equations included in attempt The Attempt at a Solution I was confused as to whether I include the power series for ln(x) in the solution. It makes comparing coefficients very nasty though. Whenever I expand for m=0 for the a0 I end...

This equation is separable... $\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x}&= 3\,\sqrt{4 - y^2} \\ \frac{1}{\sqrt{4 - y^2}}\,\frac{\mathrm{d}y}{\mathrm{d}x} &= 3 \\ \int{ \frac{1}{\sqrt{4 - y^2}}\,\frac{\mathrm{d}y}{\mathrm{d}x} \,\mathrm{d}x} &= \int{ 3\,\mathrm{d}x} \\ \int{...

What exactly is the difference between a virtual displacement and a differential displacement? It seems like they are really the same thing if by differential we assume we are talking about a distance whose magnitude approaches zero.

Homework Statement 2. Homework Equations 3. The Attempt at a Solution [/B]- So with the (from what i interpret of the notes this is needed) the same boundary conditions when time is fixed, we can relate the 'fundamental problem'- the initial condition ##t=0## given by a delta...

Say you have a log-level regression as follows: $$\log Y = \beta_0 + \beta_1 X_1 + \beta_1 X_2 + \ldots + \beta_n X_n$$ We're trying come up with a meaningful interpretation for changes Y due to a change in some Xk. If we take the partial derivative with respect to Xk. we end up with...

Let x(t) a positive function satisfied the following differential inequality $\frac{x'(t)}{1+{x(t)}^{2}}+x(t)f(t)<2f(t)$ , (1) with $0\leq t\leq T$ , $\arctan(0)<\frac{\pi }{2}$ and $f(t)$ is a positive function. Is x(t) bounded for all $T\geq 0$?

It seems there must be a way, but I cannot seem to wrap my head around it. Here's the scenario... I have a 24" line flowing with a wet gas (combustion flue gas) into a dryer. It comes in at 90 degrees F, at a rate of 180 SCF/M. It leaves the dryer at 73 degrees F at a rate of 124...

Hi! I was wondering how I could come up with a differential equation for projectile motion on a 2D plane when air resistance is not negligible. I'm trying to guess the position of a projected ball at a certain time period by approximating the coordinates using the Euler's method. Here, I would...

Homework Statement I have to calculate the stationary field inside a room. Homework EquationsThe Attempt at a Solution I used the diffusion equation to calculate the temperature, which is T(x,y)=(Eeknx+Fe-knx)cos(kny), k=(n*pi/a), a is the length of the room. Now i have to satisfy boundary...

Homework Statement A rocket sled moves along a horizontal plane, and is retarded by a friction force friction = μW, where μ is constant and W is the weight of the sled. The sled’s initial mass is M, and its rocket engine expels mass at constant rate dM/dt ≡ γ; the expelled mass has constant...

I am taking my first graduate math course and I am not really familiar with the thought process. My professor told us to think about how to prove that the differential map (pushforward) is well-defined. The map $$f:M\rightarrow N$$ is a smooth map, where ##M, N## are two smooth manifolds. If...

<Moderator's note: Moved from a technical forum and therefore no template.> Hi everybody I've been trying to solve this problem all the afternoon but I haven't been able to do it, I've written what I think the answers are even though I don't know if they're correct, so I've come here in order...

What is the most satisfactory explanation for guessing certain solutions to the differential equations encountered in damped & driven SHM?

Hi everybody. I'm using the Maple 13 software (in linux mint) to solve system compounded by the four below differential equations: > ode1 := (diff(m1(t), t)) = - m1(t) + (1/2)*tanh( m2(t) + m4(t) + cos(t) ); > ode2 := (diff(m2(t), t)) = - m2(t) + (1/2)*tanh( m1(t) + cos(t) ); > ode3 :=...

My goal is to do research in Machine Learning (ML) and Reinforcement Learning (RL) in particular. The problem with my field is that it's hugely multidisciplinary and it's not entirely clear what one should study on the mathematical side apart from multivariable calculus, linear algebra...

Homework Statement A simple pendulum in quiet is released from the horizontal (##\theta=0##) (##\theta=90## for the vertical). Will the pendulum cover in the smaller time the arch from ##\theta=0## to ##\theta=30## or from ##\theta=30## to ##\theta=90##? The Attempt at a Solution I would like...

How can we work out all the properties of wave from differential equation? And what really does differential equation of wave implies?

What is a finite difference discretization for the fourth-order partial differential terms \frac{\partial u}{\partial x}k\frac{\partial u}{\partial x}\frac{\partial u}{\partial x}k(x,y)\frac{\partial u}{\partial x} and \frac{\partial u}{\partial x}k(x,y) \frac{\partial u}{\partial y}...

Homework Statement Suppose at time zero, the bob was drawn upward four units from the equilibrium position, let C=2, K=2, m=1 lbm, initial speed=2 unit/sec find an expression for body's position. and in the solution it says: y''+6y'+5y=0 my question is: from where does the numbers (6) and (5)...

This question involves finding the transfer function for the system, but I first need to get the differential equations correct. Have I set up the gearbox correctly?

Homework Statement am trying to solve this PDE (as in the attached picture) https://i.imgur.com/JDSY4HA.jpg also my attempt is included, but i stopped in step, can you help me with it? appreciated, Homework EquationsThe Attempt at a Solution my attempt is the same as in the attached picture...

Hi. I was wondering if it is possible to apply separation of variables for a function of space and time obeying a non homogeneous differential equation. In particular, the heat equation: ##\displaystyle \frac{\partial \Phi(\mathbf{r},t)}{\partial t}-\nabla \cdot \left [ \kappa(\mathbf{r})...

In undergraduate dynamics, they do things like this: -------------------- v = ds/dt a = dv/dt Then, from this, they construct: a ds = v dv And they use that to solve some problems. -------------------- Now I have read that it is NOT wise to treat the derivative like a fraction: it obliterates...

Hi everybody. I need to learn how to solve this kind of equation by decomposing y in a serie of functions. All the examples I have seen are of homogeneous functions. I would be extremely thankfull if someone pointed me to some text in which this is done-explained. Thanks for reading.

When we take a differential element for analysis why don't we consider quantum effects and only consider classical mechanics to solve the problem?

Homework Statement How does one show that q(t) is indeed a solution? Homework EquationsThe Attempt at a Solution My current idea is that i should come up with any form of solution, like q = Acos(ωt), and slot it in the RHS. Reason being that if q is indeed a solution, the result of the...

Hello! Can someone point me toward some (introductory) problems in differential geometry with solutions (preferably free)? Thank you!

Homework Statement Homework Equations General wave solution y=f(x+ct)+g(x-ct) [/B] The Attempt at a Solution [/B] Graphical sketch

If you consider b^2/m > 4*k, you can get the solution by using classic method (b = damping constant, m = mass and k = spring constant) otherwise you have to use complex numbers. How have the references books proved the solution for this differential equation?

Homework Statement We can treat the following coupled system of differential equations as an eigenvalue problem: ## 2 \frac{dy_1}{dt} = 2f_1 - 3y_1 + y_2 ## ## 2\frac{dy_2}{dt} = 2f_2 + y_1 -3y_2 ## ## \frac{dy_3}{dt} = f_3 - 4y_3 ## where f1, f2 and f3 is a set of time-dependent sources, and...

Homework Statement Two ideal van der Waals fluids are contained in a cylinder, separated by an internal moveable piston. There is one mole of each fluid, and the two fluids have the same values of the van der Waals constants b and c; the respective values of the van der Waals constant ''a'' are...

Homework Statement Find all solutions of the given differential equations: ## \frac{dx}{dt} = \begin{bmatrix} 6 & -3 \\ 2 & 1 \end{bmatrix} x ## Homework EquationsThe Attempt at a Solution So, we just take the determinate of A-I##\lambda## and set it equal to 0 to get the eigenvalues of 3...

Exact differential of a scalar function f takes the form of ∇f⋅dr=Σ∂ifdxi (where dr is a vector) f:R->Rnand I am not sure why this equation is valid in the sense that if we integrate the equation, ∫∇f⋅dr=∫{Σ∂ifdxi} ∫df=∫{Σ∂ifdxi} the above equation is true because integration is a linear...

I'll cut the long story short. What on Earth happened here: I seem to be unable to do the integration by parts of the first term. I end up with a lot of dx's.

Homework Statement A cantilever of length ##L## is rigidly fixed at one end and is horizontal in the unstrainted position. If a load is added at the free end of the beam, the downward deflection, ##y##, at a distance, ##x##, along the beam satisfies the differential equation...

Firstly I know how to do this with first derivatives in differential equations - for example say we had ##\frac{dy}{dx}=4y^2-y##, and we're also told that ##y=1## when ##x=0##. ##\frac{dy}{dx}=4y^2-y## ##\frac{dx}{dy}=\frac{1}{4y^2-y}=\frac{1}{y\left(4y-1\right)}=\frac{4}{4y-1}-\frac{1}{y}##...

Hi, I working on their text this equation did not make sense to me. From equation 1 it differentiate second term , I wonder how he got second term of equation 2. What I think is, what I wrote at the bottom

Hello everybody. Consider $$\frac{\partial}{\partial t}f(x) + ax\frac{\partial }{\partial x}f(x) = b x^2\frac{\partial^2}{\partial x^2}f(x)$$ This is the equation (19) of...

Homework Statement I was reading a PDE book with a problem of resonance $$ y_{tt} (x,t) = y_{xx} (x,t) + A \sin( \omega t) $$ After some work it arrived to a problem of variation of parameters for each odd eigenvalue. To solve it, it uses $$ y''(t)+a^{2} y(t) = b \sin ( \omega t) \qquad y(0)=0...

Hi, I need to solve a system of first order partial differential equations with complex variables given by What software should I use for solving this problem..? The system includes 13 differential equations ...

Homework Statement A particle of mass m is subject to a force F (x) = -kx. The initial position is zero, and the initial speed is v0. Find x(t). Homework Equations F = m*v*dv/dx = -kx v = dx/dt The Attempt at a Solution I'm new to differential equations, so please excuse me if I make any...

I have a physics project at university, we designed an experiment to measure the effectiveness of Poiseuilles law in a 'quasi non-steady state'. Poiseuilles law, simply being the measurement of the flow rate of a fluid in a pipe, holding only under steady state though. So by quasi steady state I...

I've run across several instances while doing homework where a question will have two solutions. One will be an equation, and the 2nd will be a constant (usually zero). I can't figure out why this constant is a solution though. For example, take the following differential equation...

Homework Statement [/B] Suppose that $$xf(x,y)dx+yg(x,y)dy=0$$ Solve: $$f(x,y)dx+g(x,y)dy=0$$ Homework EquationsThe Attempt at a Solution Well, I'm mostly stumbling around in the dark. I tried a few things and got nowhere before heading down this road. First I solved for ##f(x,y)dx## in the...

I'm helping some guys with Calculus I class and found this exercise in the practice about integrals. I think it's overkill but it may have some easy way to solve it. I'm very rusty solving differential equations. 1. Homework Statement Find f differentiable such that $$ (3+f'(x))e^{2-x} = (x-6)...

Homework Statement Approximate ##~\sqrt[4]{17}~## by use of differential Homework Equations Differential: ##~dy=f(x)~dx## The Attempt at a Solution $$y=\sqrt[4]{x},~~dy=\frac{1}{4}x^{-3/4}=\frac{1}{4\sqrt[4]{x^3}}$$ $$\sqrt[4]{16}=2,~~dx=1,~~dy=\frac{1}{4\sqrt[4]{x^2}}\cdot 1=0.149$$...

Homework Statement Hi, I am looking at this question: With this (part of ) solution: Homework EquationsThe Attempt at a Solution I follow up to the last line- I do not understand here how we have simply taken the ##1/t^{\alpha m + \alpha}## outside of the derivative...

Hello! I would like to know if anybody here knows if there's any good book on academic-level dfferential geometry(of curves and surfaces preferably) that emphasizes on geometrical intuition(visualization)? For example, it would be great to have a technical textbook that explains the geometrical...

If the slope of the curve (derivative) at a given point is a number .