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. C

    A Partial differential equation containing the Inverse Laplacian Operator

    I am trying to reproduce the results of a thesis that is 22 years old and I'm a bit stuck at solving the differential equations. Let's say you have the following equation $$\frac{\partial{\phi}}{\partial{t}}=f(\phi(r))\frac{{\nabla_x}^2{\nabla_y}^2}{{\nabla}^2}g(\phi(r))$$ where ##\phi,g,f## are...
  2. sophiatev

    I Invariance of Diff. Line Elems. in Hartle's Gravity: Intro to GR

    In Hartle's book Gravity: An Introduction to Einstein's General Relativity he spends chapter 2 discussing some basic aspects of differential geometry. For example, he derives the expression for a differential line element in 2D Euclidean space: dS^2 = (dx)^2 + (dy)^2 in Cartesian coordinates...
  3. snoopies622

    Arriving at the differential forms of Maxwell's equations

    In college I learned Maxwell's equations in the integral form, and I've never been perfectly clear on where the differential forms came from. For example, using \int _{S} and \int _{V} as surface and volume integrals respectively and \Sigma q as the total charge enclosed in the given...
  4. Ayoub Tamin

    Mathematica How to solve this differential equation using Mathematica's Dsolve?

  5. K

    B Differential equations in physics

    Can someone list to me (and whoever is going to view this thread) what topics in differential equations should be studied so that we can have a decent knowledge of the general physical theories in which they occur? (And I believe, they appear in all theories.) So far, I believe the two most...
  6. D

    I Help getting started with this differential equation

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  7. karush

    MHB Apc.9.3.1 solution to the differential equation condition

    253 Which of the following is the solution to the differential equation condition $$\dfrac{dy}{dx}=2\sin x$$ with the initial condition $$y(\pi)=1$$ a. $y=2\cos{x}+3$ b. $y=2\cos{x}-1$ c. $y=-2\cos{x}+3$ d. $y=-2\cos{x}+1$ e. $y=-2\cos{x}-1$ integrate $y=\displaystyle\int 2\sin...
  8. A

    Differential Equations and Damper Curves

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  9. T

    Variations of a parameter in a differential equation

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  10. K

    A Differential movement along a curved surface

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  11. V

    Courses Is it Worth Retaking a "C" in Differential Equations?

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  12. C

    Engineering Differential Amplifier with an Active Load

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  13. D

    I Differential Geometry: Comparing Metric Tensors

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  14. Anonymous_

    I What is the most useful method to solve differential equations?

    What is yours?
  15. chaksome

    Solution for a second-order differential equation

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  16. chwala

    Solve the differential equation

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  17. anooja559

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  18. B

    First-order nonlinear differential equation

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  19. J

    Stirling engine LTD (Low Temperature Differential) plate size / energy

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  20. Z

    I Area Differential in Cartesian and Polar Coordinates

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  21. A

    Partial Differential Equations result -- How to simplify trig series?

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  22. giveortake

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  23. J

    I Solving Differential Equations in General Relativity

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  24. K

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  25. A

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  26. kepherax

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  27. S

    Geometry Differential Geometry: Book on its applications?

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  28. T

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  29. Terrycho

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  30. D

    Ampere's law in differential form

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  31. S

    Solving this partial differential equation

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  32. Sara_76

    B Solve a second-order differential equation

  33. P

    How do I solve these differential equations?

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  34. Abhishek11235

    I Condition for delta operator and total time differential to commute

    While deriving continuity equation in Fluid mechanics, our professor switched the order of taking total time derivative and then applying delta operator to the function without stating any condition to do so(Of course I know it is Physics which alows you to do so) . So,I began to think...
  35. W

    I How can we identify non-linear singular differential equation

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  36. endykami

    Differential calculus, solve for y: 4(y''y'')+(y'y')-1=0

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  37. B

    Solve the differential equation: y′′y′+yy′+yy′′=0

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  38. J

    MHB Differential equation - distance needed to achieve target speed

  39. A

    I Maxwell's equations with differential forms

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  40. S

    Set up the differential equation showing the voltage V(t) for this RC circuit

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  41. K

    I Find Practical Resonance Frequencies in Linear Differential Equations

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  42. murshid_islam

    Differential equation problem: Solve dy/dx = (y^2 - 1)/(x^2 - 1), y(2) = 2

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  43. Kaguro

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  44. Hiero

    I Vector valued integrals in the theory of differential forms

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  45. B

    A Making sense of the Differential of F at p, (where F: R^n -> R^m)

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  46. B

    A I need some fun questions with answers in differential geometry ()

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  47. C

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  48. PainterGuy

    I Solving a differential equation using Laplace transform

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  49. karush

    MHB -m30b Convert the differential equation

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  50. L

    I Understanding vector differential

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