What is Differential equations: Definition and 999 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. D

    Green's functions for translationally invariant systems

    As I understand it a Green's function ##G(x,y)## for a translationally invariant differential equation satisfies $$G(x+a,y+a)=G(x,y)\qquad\Rightarrow\qquad G(x,y)=G(x-y)$$ (where ##a## is an arbitrary constant shift.) My question is, given such a translationally invariant system, how does one...
  2. L

    SHM - as two ordinary linear differential equations

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  3. Hutchyy

    Ordinary Differential equations question

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  4. PhotonSSBM

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  5. ecoo

    Integrating differential equations that have ln

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  6. Julio1

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  7. S

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  8. Mark Brewer

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  9. M

    Differential equations: Circuit RL

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  10. S

    Learning Differential equations, week 2 level material

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  11. D

    Third Order Differential Equations (Mass-Spring-Damper-etc)

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  12. Q

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  13. D

    System of ODE - comparison with paper

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  14. T

    Zero-Input/Zero-State Response vs. Homogenous/Particular Solution

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  15. VoteSaxon

    Finding the Particular Integral for d2y/dt2 + 4y = 5sin2t

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  16. Mark Brewer

    Ordinary Differential Equations (ODE) Problem

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  17. H

    Seek power series solutions of the given differential equation

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  18. T

    Ordinary Differential Equations - Existence/Uniqueness Proof

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  19. J

    Setting up differential equations for voltage

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  20. Luterinho

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  21. kostoglotov

    Will linear algebra help me in more advanced diff eqs study?

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  22. kostoglotov

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  23. K

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  24. kostoglotov

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  25. L

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  26. MexChemE

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  27. Lagraaaange

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  28. L

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  29. kostoglotov

    Another Diff Eq's one: am I wrong?

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  30. kostoglotov

    Diff Eqs: can't get a reasonable answer

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  31. Destroxia

    Undetermined Coefficients, Differential Equations

    Homework Statement y'' + y =3*sin(2t) +t*cos(2t) Okay, so I have found the complimentary solution, and the first partial solution as listed in my work below. My problem is the work on the second partial solution. I have got all the derivatives plugged into the differential equation, my...
  32. M

    MHB Differential equations - Decidability and Complexity

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  33. T

    MHB Differential Equations of Calculus Problems- Calc I course

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  34. Destroxia

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  35. Destroxia

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  36. Destroxia

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  37. H

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  38. S

    Differential equations [arctan(x) to tan(x)]

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  39. L

    Frequency of a wire/spring system?

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  40. H

    System of nonlinear differential equations

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  41. J

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  42. SrVishi

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  43. S

    Coupled differential equations for charged particles

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  44. B

    Two differential equations -- Need to find steady state values

    Homework Statement I have found the two equations describing the system. They are. ċ(t) = 1/θ [ f`(k(t)) -(n + δ + β)]c(t) k̇(t) =f(k(t) - (n +d)k(t) - c(t) Plugging in the numbers: ċ(t) = 2 [ 0,25k^(-0,25) -(0,10)]c(t) k̇(t) =k^0,25 - (0,6)k(t) - c(t) Since ċ(t) and k̇(t) =0 in steady...
  45. Prof. 27

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  46. SU403RUNFAST

    Differential equations, orthogonal trajectories

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  47. twoski

    Differential Equations y" + y = tanx, solve this DE

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  48. Destroxia

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  49. mudweez0009

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  50. I

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