What is Differential: Definition and 1000 Discussions
In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.
Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.
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.
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...
ฮด
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: ๐ฟ๐ด = ๐๐๐ โ ๐๐ = ๐(๐๐) โ ๐๐๐ โ ๐๐ = โ๐(๐ โ ๐๐) โ ๐๐๐ = โ๐๐น โ ๐๐๐
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...
##ฮท=-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 {ฯ}{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...
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...
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
$$
f(z) = e^{-z^2} \big( c_1 + c_2 \Phi(\sqrt{3}z) \big)
\quad \textnormal{($\Phi(x)$ is probit function.)}...
Hi, I'm differentiating the "z" function to find extreme points but after solving the first partial derivatives with respect to "x" and "y" and also the "x" variable of the system, I can't find "y" (still in the system) using "ln" (natural logarithm).
The question is can I differentiate both...
I was thinking of using the chain rule with
dF/dx = 0i + (3xsin(3x) - cos(3x))j
and
dF/dy = 0i + 0j
but dF/dy is still a vector so how can it be inverted to get dy/dF ?
what are the other methods to calculate this?
The original differential equation is:
My solution is below, where C and D are constants. I have verified that it satisfies the original DE.
When I apply the first boundary condition, I obtain that , but I'm unsure where to go from there to apply the second boundary condition. I know that I...
ok I posted this a few years ago but replies said there was multiplication in it so I think its a mater of format
##\dfrac{\partial u^2}{\partial x\partial y}## is equivalent to ##u_{xy}##
textbook
Hi, last semester I "solved" a full differential equation and the answer was (see the picture). What does it mean? Can I make a graphic with it or what? I really don't get it.
*Arrows are just a continuation of the main formula*
I am currently looking at section IIA of the following paper: https://arxiv.org/pdf/gr-qc/0511111.pdf. Eq. (2.5) proposes an ansatz to solve the spheroidal wave equation (2.1). This equation is
$$ \dfrac{d}{dx} \left((1-x^2) \dfrac{d}{dx}S_{lm} \right) + \left(c^2x^2 + A_{lm} -...
I ordered Differential Equations and Boundary Value Problem ( Computing and Modelling) by Edwards and Penney. There are several things in the book which I don't like
Too much focus is given to modelling, almost every topic is explained not from mathematical point of view but from application...
[Mentor Note -- This thread start is by a new member from the recent MHB forum merger]
Hello Guys, I want to find a friend with whom I can discuss differential equations!
I would like to do that via WhatsApp or zoom applications.
I am interested in applications of differential equations ( for...
I am currently reading about small signal analysis of a MOSFET differential amplifier. The text I am using has the below two figures for a common-mode equivalent circuit for the amplifier. The first makes sense to me except where it calls it a "Norton equivalent circuit", whereas I thought a...
Is there a good rubric on how to choose the order of polynomial basis in an Finite element method, let's say generic FEM, and the order of the differential equation? For example, I have the following equation to be solved
## \frac{\partial }{\partial x} \left ( \epsilon \frac{\partial u_{x}...
Hello, there. I am trying to solve the differential equation, ##[A(t)+B(t) \partial_t]\left | \psi \right >=0 ##. However, ##A(t)## and ##B(t)## can not be simultaneous diagonalized. I do not know is there any method that can apprixmately solve the equation.
I suppose I could write the...
Hi,
The following circuit is given, where the switch S is closed at time t=0.
a) Set up the general differential equation (DE) for the current i(t) and bring the result into the following form## \frac{di(t)}{dt} +c_1 i(t)=c_0,## with the constant terms c0 and c1.
Hint: Determine the DE using...
Hello everyone. I hope anyone can help me with this problem. I will greatly appreciate it. Willing to compensate anybody to answer this problem correctly for me.
Hi ... I have written the equation of family of straight lines which are tangent to the circle as :
y=(-m/n)x+(m^2/n)+n
line intersects circle at : (m,n)
But I can't understand how to find differential equation of this ...
I will be appreciated if anyone has extra time to give me a little...
Hello!
Disclaimer: I am not really sure in which forum I should post this problem since the homework is electrical engineering,but the problem I am facing is of mathematical nature (at least I think).
Consider this circuit;
The given RC network contains the resistors R1 = 200 ฮฉ and R2 = 300...
Can someone help me understand the answer to this differential?
I have the following expression
where
Now what I can understand the differential of
what will be the following?
If the right-hand side is zero, then it will be a wave equation, which can be easily solved. The right-hand side term looks like a forced-oscillation term. However, I only know how to solve a forced oscillation system in one dimension. I do not know how to tackle it in two dimensions.
I have...
My question i am trying to solve:
I have successfully done first order equations before but this one has got me a little stuck. My attempt at the general solution below:
$${5} \frac{\text{d}\theta}{\text{d}t}=-6\theta$$
$${5} \frac{\text{d}\theta}{\text{d}t} =\frac{\text{-6}\theta}{5}$$...
The solution in my book:
5/4 = 1.25. That is 25 % more.
What I came up with:
I thought that now we have totally 9 players. So A: 4/9 and B: 5/9. The difference is 1/9 which is about 11%!
A friend told me :
The difference between B & A is 5-4=1
The changing rate is (5-4)/5 = 0.2 !
So B has 20...
I am going through this page again...just out of curiosity, how did they arrive at the given transforms?, ...i think i get it...very confusing...
in general,
##U_{xx} = ฮพ_{xx} =ฮพ_{x}ฮพ_{x}= ฮพ^2_{x}## . Also we may have
##U_{xy} =ฮพ_{xy} =ฮพ_{x}ฮพ_{y}.## the other transforms follow in a similar manner.
Hello,
I know that this question might be a bit silly but I am confused about plotting a normalized differential cross section. Suppose that I have a histogram with the x-axis representing some observable X and the y-axis the number of events per bin. I want the y-axis to show the normalized...
*** MENTOR NOTE: This thread was moved from another forum to this forum hence no homework template.
Summary:: Trying to find transfer functions to design a block diagram on simulink with a PID controller and transfer functions for a water tank system.
----EDIT---
The variables and parameters...
Summary:: Differential equation of motion, parabola
Hi. I've tried resolve this problem but I have two doubts. The first is about the differential equation of motion because I can't simplify it to the form y" + a*y' + b*y = F(t). I'm not sure if what I got is right. My second doubt is that I...
I'd like a good set of notes or a textbook recommendation on how to approach vector differential equations. I'm looking for something that isn't specific to one type of application like E&M, fluid dynamics, etc., but draws heavily from those and other fields for examples.
I'd strongly prefer a...