What is Differential equation: 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. Cocoleia

    A Wronskian- variation of Params Problem

    Homework Statement y''-4y'+4y=(12e^2x)/(x^4) I am trying to solve this differential equation. I know you would use the variation of parameters method, and I am trouble with the wronskian. My solution manual doesn't actually use a wronskian so I can't verify my work Homework EquationsThe...
  2. T

    A How is this 'root stability' differential equation derived?

    I'm currently studying the sensitivity of polynomial roots as a function of coefficient errors. Essentially, small coefficient errors of high order polynomials can lead to dramatic errors in root locations. Referring to the Wilkinson polynomial wikipedia page right...
  3. G

    Modeling epidemics - solving differential equation

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  4. K

    Modeling differential equation

    Homework Statement Liquid is pouring into a container at a constant rate of 30cm^3s^-1 At time t seconds liquid is leaking from the container at a rate of 2/15 V cm^3s^-1, where V cm^3 is the volume of liquid in the container at that time. Show that -15 dV/dt = 2V - 450Homework Equations ...
  5. M

    A Nonlinear differential equation

    ρCp (∂T/∂t) + k (∂2T/∂x2) = exp(-σt2)exp(-λx2)φo i have this equation... i was thinking of taylor series expansion to solve it and make it easier... any ideas on how to solve?
  6. dykuma

    Partial Differential equation, Temp in a Cylinder

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  7. G

    Finding the distance traveled by a steamer

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  8. L

    A Please help with 2-variable partial differential equation

    As a part of my research work, I need to find the number of charged particles at a given time 't', at a distance 'x' from anode. I derived a set of PDEs as per my requirement and assumptions which needs to be solved analytically. \begin{equation} \frac{\partial{N_e}}{\partial{t}} = \alpha N_e...
  9. T

    I Derivative of A Def. Integral Equals Another Def. Integral?

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  10. M

    Obtaining General Solution of ODE

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  11. dykuma

    How Does Viscosity Affect Motion in a Fluid?

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  12. Dyatlov

    I Second order differential equation

    I am trying to solve this equation: d/dx[dF(x)/dx] = [c(c+1)/x^2)F(x), where c is a constant. Do I still use the characteristic equation to solve this? EDIT: Is it solvable using Dawson's integral rule?
  13. Kanashii

    Solve for the solution of the differential equation

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  14. S

    Find the exact value of a differential equation.

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  15. T

    I Legendre Differential Equation

    I just started learning Legendre Differential Equation. From what I learn the solutions to it is the Legendre polynomial. For the legendre DE, what is the l in it? Is it like a variable like y and x, just a different variable instead? Legendre Differential Equation: $$(1-x^2) \frac{d^2y}{dx^2}...
  16. K

    A Need help with a differential equation

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  17. samg1

    Ordinary Differential Equation - tank with inflow and outflo

    Homework Statement A tank contains 60 kg of salt and 2000 L of water. A solution of a concentration 0.015 kg of salt per liter enters a tank at the rate 6 L/min. The solution is mixed and drains from the tank at the same rate. Find the amount of salt in kg at t = 3 hours Find the...
  18. Dusty912

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  19. K

    B Private solution to a polynomial differential equation

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  20. L

    I C1 and C2 in a differential equation....

    d^2x/dt=0.01-0.01dx/dt =>x(t)=-100c1(e^-0.01t)+c2+t How do we find c1 and c2. Are they numbers or functions? d^2x/dt^2 instead of d^2x/dt gives the same solution, which means different c1 and c2
  21. Dusty912

    Separable Differential Equation

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  22. Dusty912

    Separable differential equation

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  23. Elvis 123456789

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  24. O

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  25. A

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  26. D

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  27. M

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  28. Elvis 123456789

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  29. J

    A Second Order Differential equation Bessel-type

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  30. E

    A Triple Product in Laplace Transform

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  31. nysnacc

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  32. Mr Davis 97

    Differential equation for air resistance

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  33. Ian Baughman

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  34. R

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  35. B

    Confused on simple differential equation

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  36. S

    Solution to Differential Equation: Particle Motion with Initial Velocity of 2k

    Homework Statement The equation of motion of a particle is given by the differential equation ##\frac{d^2x}{dt^2} = -kx##, where ##x## is the displacement of the particle from the origin at time ##t##, and ##k## is a positive constant. 1. Show that ##x = A\cos{(kt)}+B\sin{(kt)}##, where ##A##...
  37. J

    Differential equation: solving for the General Solution

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  38. J

    A How to find a differential equation if I know its solution?

    Hello folks, Let me explain this. Much effort, time, methods and books is devoted to the science of finding solutions of differential equations. But I cannot find anywhere the reverse problem. Basically I am thinking about the work that Maxwell did finding the differential equations that...
  39. J

    Matrix riccati differential equation using matlab

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  40. I

    B Can you reason out a differential equation using logic?

    Let us suppose that we are modeling a particle traversing some distance in space & that f(x)=x². Let the x-axis be time (hrs) & the y-axis be position in space (miles). What is the instantaneous velocity of said particle at t=4 hours? Let's assign variables to each coordinate in our x, y...
  41. petterson

    A Differential equation with integral

    Hi, I was wondering if anyone had some advice on how to solve the following equation for ## F(b)##: $$ F(g(b)) h(b) + F'(b) s(b) - F(b)h(b) + h(b) + \int_{g(b)}^{b} v(x) F'(x) dx = 0 $$ Any hints on how to tackle this would be highly appreciated. Thank you!
  42. P

    Modeling survival with a differential equation

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  43. B

    I How can I verify the solution for Poisson equation with forgotten factor?

    Hello I have a physics article who solve Poisson equation of the form: L2 ∂2f/∂x2=sinh(f) The proposed solution is: tanh(f/4)=exp(x/L) tanh(f0/4) with f0 a constant I suspect an error, something like a forgotten factor. How can I verify? (I tried but I failed) I forgot the limit...
  44. chwala

    Solving a differential equation using integrating factor

    Homework Statement Solve ##{dy/dx}-2xy=2x##Homework EquationsThe Attempt at a Solution Let ##P= -2x ## and Q= 2x, Integrating factor =## e^{-x^2} ## ##y.e^{-x^2} = ∫ 2x.e^{-x^2} dx## ##y.e^{-x^2}={x^2} e^{-x^2}+∫ 2{x^3} e^{-x^2}dx## since ##y.e^{-x^2} = ∫ 2x.e^{-x^2} dx## then...
  45. awholenumber

    B A few differential equation questions

    i was trying to learn differential equations ... i was lucky enough to find some good explanations for calculus and differential equations ... Mod note: unrelated image now deleted differential equations in this equation which is the dependent variables and which one is the independent...
  46. S

    I Help with basic problem of kinematics

    Well, the problem says: From some height a object with mass m is thrown . Determinate the law that describes how the velocity of fall v changes, if on the object, besides gravity, acts the force air resistance, which is proportional to velocity v (the proportionality coefficient is k), ie must...
  47. AwesomeTrains

    Green's function differential equation

    Hello I'm doing some problems in QM scattering regarding the Green's function. Homework Statement Determine the differential equation of G(\vec{r},\vec{r}',\omega) Homework Equations I've been given the Fourier transform for the case where the Hamiltonian is time independent...
  48. P

    Integrating a Tricky Differential Equation with a Square Root Fraction

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  49. goonking

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  50. Summer95

    I Confused about Finding the Green Function

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