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. A

    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...
  2. E

    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.
  3. chwala

    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}##...
  4. MatinSAR

    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...
  5. A

    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 Ω and R2 = 300...
  6. ssafdarpk

    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?
  7. Haorong Wu

    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...
  8. B

    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: $${5} \frac{\text{d}\theta}{\text{d}t}=-6\theta$$ $${5} \frac{\text{d}\theta}{\text{d}t} =\frac{\text{-6}\theta}{5}$$...
  9. Pouyan

    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...
  10. chwala

    Proof involving ##ω(ξ,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} = ξ_{xx} =ξ_{x}ξ_{x}= ξ^2_{x}## . Also we may have ##U_{xy} =ξ_{xy} =ξ_{x}ξ_{y}.## the other transforms follow in a similar manner.
  11. Anava1001

    A Normalized differential cross section

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  12. M

    From differential equations to transfer functions

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  13. D

    Finding the differential equation of motion

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  14. The Bill

    Applied Resources for general vector differential equations?

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  15. TheSpectacularSpecs

    I Surface Differential Rotation by measuring the position of sunspots

    I am doing a University lab project where I measure positions of sunspots (using images from NASA's SDO) and use them to calculate the rotation of the Sun. Currently, all is going well: I have the angular velocity of several sunspots at varying heights. However, I want to be able to find the...
  16. mcastillo356

    Foundations On how to best introduce the concept of differential in Physics

    Hi, PF Here is the text I've taken a look at file:///C:/Users/usuario/Desktop/2001_JMT_Girep.pdf And the article I'm looking for: Artigue M. and Viennot L. Some aspects of students' conceptions and difficulties about differentials, Misconceptions and Edu. Strategies in Sci&Math. Cornell...
  17. MichaelBack12

    Best way to teach myself differential forms?

    Any suggestions? Online courses or videos?
  18. S

    Classification of a second order partial differential equation

    Hello! Consider this partial differential equation $$ zu_{xx}+x^2u_{yy}+zu_{zz}+2(y-z)u_{xz}+y^3u_x-sin(xyz)u=0 $$ Now I've got the solution and I have a few questions regarding how we get there. Now we've always done it like this.We built the matrix and then find the eigenvalues. And here is...
  19. A

    Domain of definition differential equations

    Hello. Considering this DE; $$ x^7 x' = (x^8-300)t^6 $$ with inital value x(0) = -2 Now the solution for the initial value should be C = -44; And for x(t) I get ; $$x(t) = (-44 e^{\frac{8}{7} t^7} + 300)^{\frac{1}{8}}$$ Now to get the biggest domain of definition I did this; $$ -44...
  20. V

    I Limit cycles, differential equations and Bendixson's criterion

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  21. pellman

    I What is differential about differential forms?

    Why are n-forms called differential forms? What is differential about them? And why was the dx notation adopted for them? It must have something to do with the differential dx in calculus. But dx in calculus is an infinitesimal quantity. I don't see what n-forms have to do with infinitesimal...
  22. J

    I Integral-form change of variable in differential equation

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  23. chwala

    Solve the first order differential equation

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  24. L

    I Problem with integrating the differential equation more than once

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  25. Somaiyah

    Help with deciding electives: Differential Geometry or Quantum Info

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  26. chwala

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  27. K

    I Definition of order of a partial differential equation

    How is the order of a partial differential equation defined? This is said to be first order: ##\frac{d}{d t}\left(\frac{\partial L}{\partial s_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0## And this second order :##\frac{d}{d t}\left(\frac{\partial L}{\partial...
  28. W

    A Solving nonlinear singular differential equations

    hi, i am going through differential equations which are nonlinear and singular - like Lane-Emden equation etc. my question is how to tackle singularity - while coding. regards
  29. cianfa72

    I Gradient as vector vs differential one-form

    It seems to me there is a little of confusion about the definition of gradient. Take for instance a smooth function ##f## defined on a differentiable manifold. Which is actually its gradient at a given point ? Someone says gradient is the vector ##\nabla f## defined at each point, whilst...
  30. chwala

    Using separation of variables in solving partial differential equations

    I am reading on this part; and i realize that i get confused with the 'lettering' used... i will use my own approach because in that way i am able to work on the pde's at ease and most importantly i understand the concept on separation of variables and therefore would not want to keep on second...
  31. M

    MHB Is This Variant of the Navier-Stokes Equation Solvable?

    What the hell is this and is it solvable?
  32. cianfa72

    I Differential k-form vs (0,k) tensor field

    Hi, I would like to ask for a clarification about the difference between a differential k-form and a generic (0,k) tensor field. Take for instance a (non simple) differential 2-form defined on a 2D differential manifold with coordinates ##\{x^{\mu}\}##. It can be assigned as linear combination...
  33. B

    MHB First order differential equations

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  34. galaxywanderer

    Courses Differential Equations Are Eating Me Alive

    So I am a sophomore physics major at a university near my hometown. I have always been fascinated by the way studying physics makes me think about the world, and I have struggled with but enjoyed my other undergraduate physics and math classes. This semester, however, I am taking multivariable...
  35. Z

    How to calculate differential work done by a force in circular motion?

    My question is, is this correct, and if so, why the minus sign?
  36. KyleGranger

    How Does the Micsig DP2003 Handle High Voltage Measurements?

    Hi, I am looking at a Micsig DP2003 high voltage differential probe. It's rated for 5.6 kV differential voltage. I understand that the differential votlage is measuring across two terminals which can be floating (not referenced to GND). The specs say it is rated for 1 kV common mode voltage...
  37. A

    Solving a system of differential equations by fundamental matrix

    I am given this system of differential equations; $$ x_1'=2t^2x_1+3t^2x_2+t^5 $$ $$ x_2' =-2t^2x_1-3t^2x_2 +t^2 $$ Now the first question states the following; Find a fundamental matrix of the corresponding homogeneous system and explain exactly how you arrive at independent solutions And the...
  38. A

    Solving a first order differential equation with initial conditions

    Hello! Consider this ODE; $$ x' = sin(t) (x+2) $$ with initial conditions x(0) = 1; Now I've solved it and according to wolfram alpha it is correct (I got the homogenous and the particular solution) $$ x = c * e^{-cos(t)} -2 $$ and now I wanted to plug in the initial conditions and this is...
  39. W

    I Nonlinear differential equation

    hi, i am working on nonlinear differential equation- i know rules which decide the equation to be nonlinear - but i want an answer by which i can satisfy a lay man that why the word nonlinear is used. it is easy to explain nonlinearity in case of simple equation i.e when output is not...
  40. MichPod

    A A better notation for a differential?

    A ##\frac{df}{dx}## notation is problematic. Obviously, the letter 'd' has very different meaning when applied to the function or to the argument. Additionally, a separate letter '##\partial##' is used to denote a partial differential (a very rare case in math when a notation used for a general...
  41. A

    Solving a mixing problem with a differential equation

    Hello! First I tried modelling it like most mixing problems. $$ \frac{dA}{dt} = rate coming in - rate coming out $$ where dA is the volume and dt is the time rate coming in/out can be describe as; contrencation * flow rate. Now if we plug that all on $$ \frac{dA}{dt} = 35 * 0 -...
  42. M

    Reduction of Order Problem for Differential Equations Class

    Problem statement: Second order linear differential equation in standard from Reasoning:
  43. J

    Solving this Differential Equation using Convolution

    $s=c_1*\exp(-c_2*|(t)|)*r(t)$ But how can I solve $c_1+c_2$ ?
  44. J

    Is the magnetic field B→. a state function and exact differential?

    is the magnetic field B→. a state function and exact differential? I argued that it's a state function, what do you guys think
  45. Istiak

    Deriving the Differential Position Vector in Cylindrical Coordinates

    I had an equation. $$T=\frac{1}{2}m[\dot{x}^2+(r\dot{\theta})^2]$$ Then, they wrote that $$\mathrm dr=\hat r \mathrm dr + r \hat \theta \mathrm d \theta + \hat k \mathrm dz$$ I was thinking how they had derived it. The equation is looking like, they had differentiate "something". Is it just an...
  46. S

    MHB Solve Differential Eq: xe^-1/(k+e^-1) for x, k, t

    The number of organisms in a population at time t is denoted by x. Treating x as a continuous variable, the differential equation satisfied by x and t is dx/dt= xe^-1/(k+e^-1), where k is a positive constant.. Given that x =10 when t=0 solve the differential equation, obtaining a relation...
  47. L

    I Solve second order linear differential equation

    Consider the second order linear ODE with parameters ##a, b##: $$ xy'' + (b-x)y' - ay = 0 $$ By considering the series solution ##y=\sum c_mx^m##, I have obtained two solutions of the following form: $$ \begin{aligned} y_1 &= M(x, a, b) \\ y_2 &= x^{1-b}M(x, a-b+1, 2-b) \\ \end{aligned} $$...
  48. M

    I Dependency of phase space generator to differential distributions

    I attatched an example plot where I created the histogram for the differential distribution with respect to the energy of the d-quark produced in the scattering process. My conception is that the phase space generator can "decide" how much of the available energy it assigns to the respective...
  49. Leo Liu

    Gauss' law in differential form

    My book claims that the diff. form of Gauss' law is $$\nabla\cdot\mathbf E=4\pi\rho$$ Can someone tell me why it isn't ##\nabla\cdot\mathbf E=\rho/\epsilon_0##?
  50. 1

    Biology Why do these biology textbooks write 'differential' as an adjective?

    Are these books using 'differential' correctly? Why not just write 'different'? 1. Handbook of the Biology of Aging edited by Edward J. Masoro, Steven N. Austad. p 480. 2. The Cambridge Companion to the Philosophy of Biology edited by David L. Hull, Michael Ruse. p 46. 3. p 78. 4. Dictionary...
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