Differential Definition and 1000 Threads

I am looking at this link; https://tutorial.math.lamar.edu/Classes/DE/RealRoots.aspx ##y{''} +11y{'} +24 y=0, y(0)=0, y^{'}(0)=-7## Now the general approach of applying boundary conditions directly is quite straightforward to me. I am interested in using an alternative approach, that is the...

This is exercise 1.8.3 from Foster & Nightingale: Show that if ##\sigma_{ab} = \sigma_{ba}## and ##\tau^{ab} =-\tau^{ba}## for all ##a##, ##b##, then ##\sigma_{ab}\tau^{ab}=0##. I began writing down ##\sigma_{ab}\tau^{ab}=\sigma_{ba}(-\tau^{ba})=-\sigma_{ba}\tau^{ba}##. Here I got stuck and...

ok, so usually one of the first equations in Diff Eq is F = -kx, which is the second order differential equation mx'' = -kx, where they give you the only general solution in the universe as Acoswt + Bsinwt. I was wondering, why can't you just separate the equation and get m∫x''/x = -∫kt'' which...

I wanted to find the differential form of the above equation and i get $$\frac{dR(t)}{dt}=R_{0}\alpha$$ (##t_{0}##=0 degree celsius) So $$\alpha=\frac{dR(t)}{dt} \frac{1}{R_{0}}$$ (##\alpha##= temperature coefficient of resistance ##R_{0}##=Resistance at temperature 0 degree celsius) This idea...

I'm trying to think of the least creepy reason I found my small lateral bathroom window closed this morning. I live in a 2 bed/1 bath apartment on the ground level with my two dogs and no one else. I rarely have people over. My bathroom doesn't have a fan and I don't have the best memory, so I...

If we have this D.E: from Latex : if I try to solve it in this way: My solution is : Which is not correct Another attempt : that gives me : What is wrong ? I know I should write: But why my integrations are wrong?

So I've been searching around for rigorous explanations for things like ##dx## in physics, I'm not looking to fully commit myself to reading the relevant literature at the moment but just want to know what I'll have to do in order to understand. Perhaps I'll make a separate thread about that...

Solve the given PDE for ##u(x,t)##; ##\dfrac{∂u}{∂t} +10 \dfrac{∂u}{∂x} + 9u = 0## ##u(x,0)= e^{-x}## ##-∞ <x<∞ , t>0## In my lines i have, ##x_t = 10## ##x(t) = 10t+a## ##a = x(t) - 10t## also, ##u(x(t),t)= u(x(0),0)e^{-9t}## note this is from, integrating ##u_t[u(x(t),t] =...

Solve the given PDE for ##u(x,t)##; ##\dfrac{∂u}{∂t} +8 \dfrac{∂u}{∂x} = 0## ##u(x,0)= \sin x## ##-∞ <x<∞ , t>0## In my working (using the method of characteristics) i have, ##x_t =8## ##x(t) = 8t + a## ##a = x(t) - 8t## being the first characteristic. For the second...

For this problem, I am trying to find the fundamental matrix, however, the eigenvalues are both imaginary and so are the eigenvectors. That is, ##\lambda_1 = 4i, \lambda_2 = -4i## ##v_1 = (1 + 2i, 2)^T## ##v_2 = (1 - 2i, 2)^T## So I think I just have an imaginary matrix? This is because the...

For this problem, Would it not make sense to use for ##x' = -sx - gx - ry## as a better version of ##x' = - gx - ry## since the ##sx## term connects the two DEs to form a coupled system (from what the author explains the ##sx## term represent insulin glucose transformation). Thanks!

For this problem, The solution is, However, can someone please explain to me where they got the orange coefficient matrix from?It seems different to the original system of the form ##\vec x' = A\vec x## which is confusing me. Thanks!

For this problem, I am confused by the term below. I get all their terms, expect replacing the highlighted term by ##e^{3t}##, does someone please know whether this is yet another typo? Thanks!

For this problem, Can someone please explain to me how they got from the orange step to the yellow step? I am confused how the two expressions are equivalent. Thanks!

Hello! I'm currently working with a problem which allows modelling ball motion $$\begin{aligned} m \ddot{x} & =-k_x \dot{x} \sqrt{\dot{x}^2+\dot{y}^2} \\ m \ddot{y} & =-k_y \dot{y} \sqrt{\dot{x}^2+\dot{y}^2}-m g \end{aligned}$$ Given that ##k_x, k_y=0.005##, ##m=0.01## and ##g=9.81## and when...

I need insight on the highlighted in Red on how ##\left[\dfrac{dz}{dx} - 1 = \dfrac{dy}{dx}\right]## otherwise the rest of the steps are clear. I just read that ##\dfrac{dx}{dy} \dfrac{dy}{dz} \dfrac{dz}{dx} =-1##

For this, I tried solving the differential equation using an alternative method. My alternative method starts at ##tv^{''} + v^{'} = 0## I substitute ##v(t) = e^{rt}## into the equation getting, ##tr^2e^{rt} + re^{rt} = 0## ##e^{rt}[tr^2 + r] = 0## ##e^{rt} = 0## or ##tr^2 + r = 0## Note that...

The PDE is $$ \frac{1}{a^2 x^2} (u_y)^2 - (u_x)^2 =1$$ I know the solution, its ## u=x senh(ay) ##, but I dont know how I can get it. I've tried variable separation and method of characteristics but they dont seem to work.

Using the concepts of Summability Calculus but generalized such that the lower bound for sums and products is also variable, we can prove that the solution to the following PDE: $$P^2\frac{\partial^2P}{\partial x\partial y}=(P^2+1)\frac{\partial P}{\partial x}\frac{\partial P}{\partial...

I'm stuck on a problem: T1 = dT2/dt + xT2 - y T2 = (Ae^(-4.26t))+(Be^(-1.82t))+39.9 I'm unsure how to proceed

Homework Statement: What actually is the particular solution of an ODE? Relevant Equations: x Consider the differential equation ##y'' + 9y = 1/2 cos(3x)##, if we wish to solve this we should first solve the auxiliary equation ##m^2 + 9 = 0## giving us ##m=3i,-3i##, this corresponds to the...

Let X be a continuous-time Markov chain that hops between two states ##\{1, 2\}## with rates ##\lambda, \mu>0##, so its generator is $$Q = \begin{pmatrix} -\mu & \mu\\ \lambda & -\lambda \end{pmatrix}.$$ Solve ##\pi Q = 0## for the stationary distribution, and verify that...

I don't understand what the question means, and the answer is provided here: https://physics.stackexchange.com/a/35821/222321 Could someone provide a comprehensive one-by-one explanation.

There are differential cross section, double differential cross section, triple DCS. What are the difference, and why is the names? Thanks.

For this, The solution is, However, why did they not move the x^2 to the left hand side to create the term ##(-2A - 1)x^2##? Is it possible to solve it this way? Many thanks!

I think my probes may not be working correctly and was seeking advice from those who know better than me. I have the Tektronics p5205 differential probes with the 1103 power supply. When I power up a test device I see a voltage waveform appear on the channel connected to the probe. The probe is...

Hello! Let $n$ be a natural number, $P_n(x)$ be a polynomial with rational coefficients, and $\deg P_n(x) = n$. Let $P_0(x)$ be a constant polynomial that is not equal to zero. We define the sequence ${P_n(x)}_{n \geq 0}$ as an Appell sequence if the following relation holds: \begin{equation}...

The time rate of change of neutron abundance ##X_n## is given by $$\frac{dX_n}{dt} = \lambda - (\lambda + \hat\lambda)X_n$$ where ##\lambda## is neutron production rate per proton and ##\hat\lambda## is neutron destruction rate per neutron. Given the values of ##\lambda## and ##\hat\lambda## at...

I will have to teach a first course in differential equations. What should I cover in this module? For example, in most books they have Laplace Transforms which is fine but I would not use LT to solve differential equations. I want to write a course that it motivates students and has an impact...

δ I had always thought that it represents a differential element for a parameter that it is not supposed to be a well-defined function - e.g., for a differential or heat or work in thermodynamics - as opposed to a regular Latin d, which is supposed to be such a well-defined function. However...

Greetings, in one of the exercise sheets we were given by our Prof, we were supposed to draw the trajectory of a patricle that moves toward a bounded spherical potential that satisfies ## V(\vec{r}) = \begin{cases} V_0 & | \vec{r} | \leq a \\ 0 & else \\ \end{cases} ## for...

I'm referring to this result: But I'm not sure what happens if I apply a linear differential operator to both sides (like a derivation ##D##) - more specifically I'm not sure at what point should each term be evaluated. Acting ##D## on both sides I'll get...

In the thermodynamics textbook there is written: 𝛿𝐴 = 𝑇𝑑𝑆 − 𝑑𝑈 = 𝑑(𝑇𝑆) − 𝑆𝑑𝑇 − 𝑑𝑈 = −𝑑(𝑈 − 𝑇𝑆) − 𝑆𝑑𝑇 = −𝑑𝐹 − 𝑆𝑑𝑇 How did we get the bolded area from TdS? Is that property of derivative, integral, or something else :/

Looking at pde today- your insight is welcome... ##η=-6x-2y## therefore, ##u(x,y)=f(-6x-2y)## applying the initial condition ##u(0,y)=\sin y##; we shall have ##\sin y = u(0,y)=f(-2y)## ##f(z)=\sin \left[\dfrac{-z}{2}\right]## ##u(x,y)=\sin \left[\dfrac{6x+2y}{2}\right]##

My thinking is two-fold, firstly, i noted that we can use separation of variables; i.e ##\dfrac{dy}{y}= \sec^2 x dx## on integrating both sides we have; ##\ln y = \tan x + k## ##y=e^{\tan x+k} ## now i got stuck here as we cannot apply the initial condition ##y(\dfrac {π}{4})=-1##...

Hello, I hope this is the right area to post this question. We are having a debate at my workplace and was hoping there was someone more qualified to settle the debate. We are a packaging company and have setup an experiment to test the pressure differnential from sealiong at 1800m vs. Sea...

I am not sure if this is correct, but here is my work by using the definition of the Gateaux differential: \begin{align*} &dS(y; \psi)=\lim_{\tau\rightarrow 0}\frac{S(y+\tau\psi)-S(y)}{\tau}=\frac{d}{d\tau}S(y+\tau\psi)\biggr\rvert_{\tau=0}\\...

I am trying to solve this homogenous linear differential equation . Since it is linear, I can use the substitution . Which gives, (line 1) (line 2) (line 3) (line 4) (line 5) Which according to Morin's equals, (line 6) However, could someone please show me steps how he got from line 5 to 6...

I was reading the oscillations chapter which was talking about how to solve linear differential equations. He was talking about how to solve the second order differential below, where a is a constant: In the textbook, he solved it using the method of substitution i.e guessing the solution...

Is it possible to apply Laplace transform to some equation of finite order, second for instance, and get the differential equation of infinite order?

This is part of the notes; My own way of thought; Given; ##U_{xy}=0## then considering ##U_x## as on ode in the ##y## variable; we integrate both sides with respect to ##y## i.e ##\dfrac{du}{dx} \int \dfrac{1}{dy} dy=\int 0 dy## this is the part i need insight...the original problem...

Here is the circuit diagram provided in the book. In the solution, the book has used the following approach (red markings in the image): Input to the left transistor is 2V. Considering base-emitter junction drop to be 0.7V, the emitters are at 1.3V (left red arrow). Now, using the "virtual...

How do we express this differential equation (dy/dx)= (y/x) + tan(y/x) into this form( Mdx + Ndy=0) where M,N are functions of (x,y) ?

I'm learning Differential Equations from Prof. Mattuck's lectures. The lectures are absolutely incredible. But there are a few topics in Tenenbaum's book and my syllabus which he doesn't seem to teach (I have reached upto lecture 14, but in future lectures too the following topics are not...

I shall not begin with expressing my annoyance at the perfect equality between the number of people studying ODE and the numbers of ways of solving the Second Order Non-homogeneous Linear Ordinary Differential Equation (I'm a little doubtful about the correct syntactical position of 'linear')...

Is there a name to this sort of differential equation? $$ f(z) + 2zf'(z) + f''(z) = 0 ~. $$ I ran into it somewhere and it does not look to be Hermite. I think it has the general solution $$ f(z) = e^{-z^2} \big( c_1 + c_2 \Phi(\sqrt{3}z) \big) \quad \textnormal{($\Phi(x)$ is probit function.)}...

Hi, I'm differentiating the "z" function to find extreme points but after solving the first partial derivatives with respect to "x" and "y" and also the "x" variable of the system, I can't find "y" (still in the system) using "ln" (natural logarithm). The question is can I differentiate both...

I was thinking of using the chain rule with dF/dx = 0i + (3xsin(3x) - cos(3x))j and dF/dy = 0i + 0j but dF/dy is still a vector so how can it be inverted to get dy/dF ? what are the other methods to calculate this?

The original differential equation is: My solution is below, where C and D are constants. I have verified that it satisfies the original DE. When I apply the first boundary condition, I obtain that , but I'm unsure where to go from there to apply the second boundary condition. I know that I...

ok I posted this a few years ago but replies said there was multiplication in it so I think its a mater of format ##\dfrac{\partial u^2}{\partial x\partial y}## is equivalent to ##u_{xy}## textbook