Differential equation Definition and 1000 Threads

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. Axel Togawa

    I Derivation of "polarization drift" in a plasma

    When studying a particle in slowly time varying, uniform electric field E, and in a constant, uniform magnetic field \textbf{B}, I found many texts where I can't understand the derivation of the "polarization drift" \textbf{v}_p, in particular I quote as reference this book I found online ([1]...
  2. Z

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  3. mathgenie

    I Taking the derivative of a function

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  4. Z

    Difference between resonance in undamped vs damped mass-spring-dashpot

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  5. Z

    Prove that solutions to autonomous first order DE can't have local maxima

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  6. M

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  7. gigorina

    I LIF Neuron Equation Solution for arbitrary time-dependent current (Neural Dynamics)

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  8. G

    How Can I Solve This Differential Equation for x(y)?

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  9. Graham87

    I Differential equation using power series method

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  10. T

    I Solutions to Simple Harmonic Motion second order differential equation

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  11. Z

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  12. M

    Determine whether ## S[y] ## has a maximum or a minimum

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  13. M

    Proofs about the second-order linear differential equation?

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  14. C

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  15. chwala

    Solve the given differential equation

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  16. chwala

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  17. U

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  18. L

    Solve these two coupled first-order differential equations and sketch the flow

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  19. L

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  20. L

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  21. M

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  22. pawlo392

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  23. gurbir_s

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  24. E

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  25. PhysicsRock

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  26. N

    Unclear differential equation from a thermodynamics textbook

    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 :/
  27. chwala

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    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]##
  28. chwala

    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 {π}{4})=-1##...
  29. M

    Solving the SHM differential equation

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  30. M

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  31. chwala

    A Solve the Partial differential equation ##U_{xy}=0##

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  32. Shovon00000

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    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) ?
  33. H

    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')...
  34. G

    I Is there a name for this sort of differential equation?

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  35. H

    Repeated roots of a characteristic equation of third order ODE

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  36. S

    Differential equation of vector field

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  37. A

    Solution to Differential Equation with Limit Boundary Condition

    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...
  38. N

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  39. dim_d00m

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    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} -...
  40. C

    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}...
  41. 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...
  42. chwala

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  43. MatinSAR

    Find the differential equation

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  44. A

    How did they get to this differential equation?

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  45. Haorong Wu

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  46. D

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  47. S

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  48. V

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  49. J

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  50. chwala

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