# 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.

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1. ### Solving a first order matrix differential equation

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...
2. ### Oscillator Differentials: What's a physical meaning of complex part of the solution for coordinate change of the anharmonic oscillator?

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.
3. ### A Differential cross section, double differential cross section, triple...

There are differential cross section, double differential cross section, triple DCS. What are the difference, and why is the names? Thanks.
4. ### Differential equation problem: y" + y' - 2y = x^2

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!
5. ### Help with Tektronix differential probes

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...
6. ### A Differential equation and Appell polynomials

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: ...
7. ### A Solving this first-order differential equation for neutron abundance

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...
8. ### What to cover in a differential equations module?

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...
9. ### I What kind of differential does the small Greek delta letter represent?

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...
10. ### Solution for differential equation

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...
11. ### I Differential operator in multivariable fundamental theorem

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...
12. ### Unclear differential equation from a thermodynamics textbook

In the thermodynamics textbook there is written: ๐ฟ๐ด = ๐๐๐ โ ๐๐ = ๐(๐๐) โ ๐๐๐ โ ๐๐ = โ๐(๐ โ ๐๐) โ ๐๐๐ = โ๐๐น โ ๐๐๐ How did we get the bolded area from TdS? Is that property of derivative, integral, or something else :/
13. ### Solve the given partial differential equation

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]##
14. ### Solve the given first order differential equation

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##...
15. ### B Investigating the Impact of Chips on Packaging Pressure Differential

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...
16. ### How to find the Gateaux differential of this functional?

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}\\...
17. ### Solving the SHM differential equation

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...
18. ### Morin classical mechanics differential equation problem

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...
19. ### A Applying the Laplace transform to solve Differential equations

Is it possible to apply Laplace transform to some equation of finite order, second for instance, and get the differential equation of infinite order?
20. ### A Solve the Partial differential equation ##U_{xy}=0##

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...
21. ### Engineering Differential amplifier confusion (BJTs + Operational Amp)

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...
22. ### I Expressing a differential equation into a different format

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) ?
23. ### Calculus Following Prof. Mattuck's lectures on Ordinary Differential Equations....

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...
24. ### I Second order non-homogeneous linear ordinary differential equation

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')...

32. ### Calculus Differential Equations book recommendations

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...
33. ### Looking for Friend to Discuss Differential Equations

[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...
34. ### Differential Amplifier Common Mode Thevenin Equivalent

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...
35. ### A FEM basis polynomial order and the differential equation order

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}...
36. ### Solutions of first-order matrix differential equations

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...
37. ### Engineering How do I find the differential equation for this circuit?

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...
38. ### MHB Oil Spill - Differential Equations

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.
39. ### Solve the problem involving differential equation

My approach for part (a), ##\dfrac{dp}{dt}+P(t)=100## I.f=## e^{\int 1 dt} = e^t## Therefore, ##(e^t p)^{'}=100e^t## ##e^tp=\int 100e^t dt## ##e^tp=100e^t+k## Applying initial condition, ##p(0)=2000## ##2000=100+k## ##k=1900## Therefore, ##e^tp=100e^t+1900## ##p=100+1900e^{-t}##...
40. ### Find the differential equation

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...
41. ### How did they get to this differential equation?

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...
42. ### A What will be value of this differential? Kindly help and explain

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?
43. ### Solving a partial differential equation

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...
44. ### Engineering First order differential equations (movement of a rotary solenoid)

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: $${5} \frac{\text{d}\theta}{\text{d}t}=-6\theta$$ $${5} \frac{\text{d}\theta}{\text{d}t} =\frac{\text{-6}\theta}{5}$$...
45. ### Interpreting Statistical Questions: Ice Hockey Player Diff.

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...
46. ### Proof involving ##ฯ(ฮพ,n)=u(x,y)## - Partial differential equations

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.
47. ### A Normalized differential cross section

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...
48. ### From differential equations to transfer functions

*** 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...
49. ### Finding the differential equation of motion

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...
50. ### Applied Resources for general vector differential equations?

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...