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. Remixex

    Question about mg on oscilations course

    On a test our teacher asked about a system composed of (string -> mass -> string -> mass) hanging, that began to oscillate up and down. We all considered weight (mg) when applying Newton's second law to find the associated differential equation. When we met our teacher again he said that we...
  2. M

    AP Physics differential equation

    Homework Statement A block of mass m, which has an initial velocity v0 at time t = 0, slides on a horizontal surface. If the sliding friction force f exerted on the block by the surface is directly proportional to its velocity (that is, f = -kv), A) Write a differential equation for the...
  3. B

    How do I solve these coupled Differential Equation?

    Homework Statement dNa/dt = -Na/Ta where Na is the function and Ta is the constant dNb/dt = Na/Ta - Nb/Tb where Nb is the function and Tb is the constant Homework Equations My Prof said Nb(t) has the form Nb(t) = Cexp(-t/Ta) + Dexp(-t/Tb) The Attempt at a Solution I know the first equation...
  4. M

    Find value of K for differential equation to be exact

    Homework Statement Hello everyone, I need to find K for the following differential equation to be exact. (y3+kxy4-2*x)dx +(3xy2 +20x2y3)dy=0 Homework EquationsThe Attempt at a Solution (y3+kxy4-2*x)dx +(3xy2 +20x2y3)dy=0 dM/dy = d/dy(y3+kxy4-2*x) = 4*k*y3*x+3*y2 dN/dx =d/dx(3xy2 +20x2y3) =...
  5. J

    Keep getting the wrong answer in this lengthy Laplace problem

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  6. M

    Homogeneous differential equation - serious help

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  7. wololo

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  8. nuuskur

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  9. R

    Engineering LC circuit (Differential Equation)

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  10. B

    Solving a problem regarding Existence theorem.

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  11. Shahrokh

    Coupled differential equation with boundary conditions

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  12. W

    First order integro differential equation

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  13. A

    Solve second order nonlinear differential equation

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  14. R

    Verifying a solution of the diffusion equation

    This is my attempt at the solution. I have been told that the given function is a solution. I just need to prove it. As you can see that I am stuck. What am I doing wrong?
  15. MidgetDwarf

    Solving a Differential Equation by Separation of Variables

    Solve the given differential equation by separation of variables. (dy/dx)= (xy+3x-y-3)/(xy-2x+4y-8) First, I noticed when i divided both sides by the left hand side and multiplied both sides by dx, nothing canceled or seemed to work. I got to thinking. on the right hand side I preformed long...
  16. H

    MHB Exponential Growth and Differential Equation

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  17. Mark Brewer

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  18. H

    Seek power series solutions of the given differential equation

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  19. W

    Is this differential equation exact?

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  20. R

    Higher order differential equation

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  21. J

    Setting up differential equations for voltage

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  22. D

    Analytical solution to mx"(t)+b(x'(t))x'(t)+k(p)x(t)=0

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  23. SSGD

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  24. kostoglotov

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  25. Destroxia

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  26. Destroxia

    Higher Order Differential Equation

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  27. Destroxia

    Finding series solution for the differential equation

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  28. Remixex

    Hooke's and Newton's law to find Second order ODE

    Homework Statement A weight of 8 pounds extends a spring 2 feet. It's assumed that the damping force that acts on the system is equal (number-wise) to alpha times the speed of the weight. Determine the value of alpha > zero so x(t) is critically damped. Determine x(t) if the weight is liberated...
  29. E

    Hyperbolic partial differential equation

    What is the general solution of the following hyperbolic partial differential equation: The head (h) at a specified distance (x) is a sort of a damping function in the form: Where, a, b, c and d are constants. And the derivatives are with respect to t (time) and x (distance). Thanks in advance.
  30. Destroxia

    Differential Equation, Substitution/Homogeneous

    Homework Statement Solve the differential equation: Here is the books answer to the problem: Homework Equations This is a substitution/homogeneous first order differential equation, which can be converted into a separable differential equation. The Attempt at a Solution Where am I...
  31. W

    Can Poker Chip Arrangements Be Optimized for Maximum Profit?

    Hey everyone. I was pondering how best to optimize a chip arrangement for a poker game. This is the scenario I've thought up: There are 4 denominations of colored chips with a set value. White (W) = 0.05 Red (R) = 0.25 Blue (B) = 1.00 Green (G) = 5.00 A player wants to purchase 40 dollars...
  32. evinda

    MHB How do we get to this differential equation?

    Hello! (Wave) If we have the initial value problem $$h''(t)=-\frac{R^2 g}{(h(t)+R)^2} \\ h(0)=0, h'(0)=V$$ we have the fundamental units: $T,L$ such that $T=[t], L=[h], L=[R], LT^{-1}=[V], LT^{-2}=[g]$ and we get the independent dimensionless quantities $\pi_1=\frac{h}{R}...
  33. evinda

    MHB General solution of differential equation

    Hello! (Wave) $$(1-x^2)y''-2xy'+p(p+1)y=0, p \in \mathbb{R} \text{ constant } \\ -1 < x<1$$ At the interval $(-1,1)$ the above differential equation can be written equivalently $$y''+p(x)y'+q(x)y=0, -1<x<1 \text{ where } \\p(x)=\frac{-2x}{1-x^2} \\ q(x)= \frac{p(p+1)}{1-x^2}$$ $p,q$ can be...
  34. LeDragonian

    How Is Displacement Solved in This Differential Equation?

    It has been a while since I was involved with my differential equations. I am a mech student. I was trying out a sample problem from the dynamics book and came upon this equation. k = some constant of proportionality for a spring pushing back a spring mounted slider. (1/k) arcsin(ks/v_0) = t...
  35. evinda

    MHB Non-linear differential equation

    Hello! (Wave) I want to find the solution $\psi$ of the non-linear differential equation $y'=1+y^2$ that satisfies the condition $\psi(0)=0$. (Notice that the solution $\psi$ exists only for $- \frac{\pi}{2}< x < \frac{\pi}{2}$) We notice that: $(tan^{-1})'(x)=\frac{1}{1+x^2} (\star) \left(...
  36. M

    Explaining a physical model (differential equation)

    Homework Statement Check on labb41.jpeg (I am not used to formatting my mathematical equations on the web, I can only format in my local text-editing programs). Homework Equations Again, check on labb41.jpeg The Attempt at a Solution I have plot the solution to U(I) as according to the...
  37. Prof. 27

    First Order Linear Differential Equation

    Homework Statement dv/dt = 9.8 - 0.196v Set in correct form: dv/dt + 0.196v = 9.8 Since p(t) = 0.196, u(t) the integration factor is given by: u(t) = e∫0.196 dt Multiply each term by u(t) and rearrange: (e∫0.196 dt)(dv/dt) + (0.196)(e∫0.196 dt)(v) = (9.8)(e∫0.196 dt) From now on we will set...
  38. D

    Differential equation of frictional force

    A question from a classical mechanics past paper described a particle of mass ##m## that had a pair of horizontal identical springs of spring constant ##k## attached on either side and that the mass is free to move horizontally. The mass is also placed on a table that gives rise to an...
  39. S

    General differential equation solution for Kepler Problem

    To be honest, I don't know any physics. I am a high school student who has taken high school physics, but America's education system isn't known for teaching much more than Newton's laws. I have, however, taken Multivariable/Vector calculus, so I have a decent math background. I was wondering...
  40. gonadas91

    Solve 1st Order ODE from Transcendental Equation

    It is a general doubt about the following equation: Imagine I want to calculate an unknown function y(x), and my starting equation is of the type y(x)^{2}=\frac{1}{x^{2}Log^{2}(A(x)y(x)^{2})} , then, am I allowed to start with the equation y(x)=\frac{1}{xLog(A(x)y(x)^{2})} and...
  41. W

    Separable partial differential equation

    Homework Statement I have two equations. cos(θ)wφ + sin(θ)wφ = 0 (1) And ## \frac{w_φ}{r}## + ∂wφ/∂r = 0 (2) Find wφ, which is a function of both r and theta. Homework EquationsThe Attempt at a Solution I end up with two equations, having integrated. wφ=## \frac{A}{sinθ}## from (1)...
  42. Prof. 27

    Linear Ordinary Differential Equation: Definition

    Homework Statement The website says this: "It is Linear when the variable (and its derivatives) has no exponent or other function put on it. So no y2, y3, √y, sin(y), ln(y) etc, just plain y (or whatever the variable is). More formally a Linear Differential Equation is in the form: dy/dx +...
  43. R

    Finding Time Taken for Particle to Travel 99 Meters Using Differential Equations

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  44. M

    Differential Equation - Series - Recurrence Relation

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  45. G

    How do I solve this stochastic differential equation?

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  46. K

    Derive gen sol of non-homogeneous DEs through linear algebra

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  47. grandpa2390

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  48. M

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  49. K

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  50. mudweez0009

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