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. Drakkith

    Differential Equation Where Y Turns Out to Equal X?

    Homework Statement [/B] Suppose that $$xf(x,y)dx+yg(x,y)dy=0$$ Solve: $$f(x,y)dx+g(x,y)dy=0$$ Homework EquationsThe Attempt at a Solution Well, I'm mostly stumbling around in the dark. I tried a few things and got nowhere before heading down this road. First I solved for ##f(x,y)dx## in the...
  2. binbagsss

    Differential Equation, Change of variables

    Homework Statement Hi, I am looking at this question: With this (part of ) solution: Homework EquationsThe Attempt at a Solution I follow up to the last line- I do not understand here how we have simply taken the ##1/t^{\alpha m + \alpha}## outside of the derivative...
  3. B

    Linear ordinary differential equation.

    Homework Statement ##\dfrac{dy}{dx} + y = f(x)## ##f(x) = \begin{cases} 2 \qquad x \in [0, 1) \\ 0 \qquad x \ge 1 \end{cases}## ##y(0) = 0## Homework EquationsThe Attempt at a Solution Integrating factor is ##e^x## ##e^x\dfrac{dy}{dx} + e^x y = e^x f(x)## ##\displaystyle ye^x = \int e^x...
  4. B

    Calculus Translation of a German book about ODEs

    I need a translation of "Differentialgleichungen : Losungsmethoden und Losugen", I guess it is written in German. This book was referenced in Shepley L. Ross' book on ODE. If the English translation is not unavailable, I am fine with a book that contains a "list" of special differential...
  5. M

    Solving an Euler differential equation

    Homework Statement Solve the differential equation ##(2x+1)^2y'' + (4x+2)y' - 4y = x^2## Can someone verify whether my solution is correct? Homework EquationsThe Attempt at a Solution We perform the substitution ##t = \ln|2x+1|##. Then, ##e^t = |2x+1|## and ##x = \pm(e^t -1)/2## Without...
  6. Polygon

    I Electrical circuit differential equation

    q''+ 20 q = 100 sin(ωt) I have been asked to find all mathematically possible values of ω for which resonance will occur. From the homogeneous solution, q(t) = Acos(√20 t) +Bsin(√20 t), I can see that resonance occurs when ω=√20. My question is, should I also consider -√20? And if so, what is...
  7. kupid

    MHB Calculus & Diff. Eqn: Beginner Qs on Function, Derivative & Gradient

    I have some beginner doubts about Calculus and Differential equations . Is a function always a curve ? Doesn't a function already has a slope ? d/dx of a function gives the gradient of the curve between two points ? The derivative ,d/dx ,The gradient , is the rate of change of a...
  8. Eclair_de_XII

    Finding a power series solution to a differential equation?

    Homework Statement "Find the recurrence relation in the power series solution for ##y''-xy'-y=0## centered about ##x_0=1##." Homework Equations ##y=\sum_{n=0}^\infty a_nx^n## Answer as given in book: ##(n+2)a_{n+2}-a_{n+1}-a_n=0## The Attempt at a Solution ##y=\sum_{n=0}^\infty a_n(x-1)^n##...
  9. Schaus

    Modeling epidemics - solving differential equation

    Homework Statement Solve for y: ##\frac {dy}{dx} = \frac {1+y^6}{xy^5}## , where y(1) = 1. Answer ## y = \sqrt[6] {2x-1}## Homework EquationsThe Attempt at a Solution ##\frac {dy}{dx} = \frac {1+y^6}{xy^5}## ##\frac{dy (y^5)}{1+y^6} = dx \frac {1}{x}## u= 1+y6 ##\frac {du}{y^5}=dx## ##\int...
  10. R

    Plotting a RLC Circuit: Analyzing a Differential Equation

    I'm not sure about the physical behavior of a RLC circuit and I have to give a presentation that involves one. So I've decided to plot the current. I found a book that gives a differential equation to describe the circuit. ##L\frac{d^2i}{dt^2} + R\frac{di}{dt} + \frac{1}{C}i = \frac{dv}{dt}##...
  11. F

    Differential equation for a pendulum

    Homework Statement A simple pendulum is formed by a light string of length ##l## and with a small bob ##B## of mass ##m## at one end. The strings hang from a fixed point at another end. The string makes an angle ##\theta## with the vertical at time ##t##. Write down an equation of motion of...
  12. D

    Inverting Shifted Laplace function

    Homework Statement A beam is supported at one end, as shown in the diagram (PROBLEM 11 page 281 of Lea, 159 of the course pack). A block of mass M and length l is placed on the beam, as shown. Write down the known conditions at x = 0. Use the Laplace transform to solve for the beam...
  13. Alettix

    Exponential Forcing Differential Equation

    Homework Statement Solve ## \frac{d^2y}{dt^2} + \omega^2y = 2te^{-t}## and find the amplitude of the resulting oscillation when ##t \rightarrow \infty ## given ##y=dy/dt=0## at ##t=0##. Homework EquationsThe Attempt at a Solution I have found the homogenious solution to be: ##y_h = A\cos\omega...
  14. O

    How to solve the differential equation for driven series RLC circuit?

    Homework Statement It is the driven series RLC circuit. It is given in the following images. It is from the section 12.3 in this note. Homework Equations The differential equation, as given by 12.3.3, is ##L \frac{d^2 Q}{d t^2} + R \frac{d Q}{d t} + \frac{Q}{C} = V_0 \sin{(\omega t)}##...
  15. Puff Cube

    Object moving upwards by constant force away from planet

    Homework Statement Suppose there is an object that is a distance ##r_0## from the center of a planet that is nearby (the object is outside the surface of the planet). Let ## r ## represent the distance from the object to the planet's center. Let ## t ## represent time. The object, which is...
  16. haushofer

    A Chaos: difference vs differential equation

    Dear all, I have a question concerning chaos. As you may well know, the logistic mapping $$x_{n+1} = rx_n (1-x_n) $$ exhibits chaos, depending on the value of r. This logistic mapping is a reparametrized version of the difference equation $$x_{n+1} = x_n + k x_n (1 - \frac{x_n}{M}) $$...
  17. T

    Second-order differential equation and conditions

    Homework Statement Hy guys I am have a problem with the last part of this question. part d), ii) I get the general formal which I have displayed below, but what I done understand is if I take the limits as show in ii) I get ##0=\ \infty## which obviously I am doing something wrong. Have I...
  18. M

    MHB Integrating Factors for Solving Linear Differential Equations

    Can i have help with this linear differential equation ? First, i divided by (1-x^2) to be like dy/dx + p(x)y= q(x). But i could not obtain Q(x). Any help will be welcomed.
  19. J

    Solving a fifth order non-homogeneous differential equation

    Homework Statement Find the general solution of y^{(5)}-y(1)=x The Attempt at a Solution I found the complementary function by substitution of the solution form y=e^{kx} giving k=0,1,-1,i,-i, so y_{cf}=a_0+a_1e^x+a_2e^{-x}+a_3e^{ix}+a_4e^{-ix} Now for the particular integral, the general...
  20. T

    Solution for a first-order differential equation

    Homework Statement determine by inspection at least two solutions of the given first-order IVP dy/dx = 3y2/3 y(0)=0 2. Equations: integral xa dx= xa+1/(a+1)+constant The Attempt at a Solution change its form to 1/y2/3 dy/dx =3 integrate both sides with respect to x then it will be 1/y2/3 dy =...
  21. Drakkith

    How Does Damping Affect the Dynamics of a Spring System?

    Homework Statement Consider the illustration of 3 springs: In A, we hang a very light spring and pan from a hinge. The pan and spring are so light, we can neglect any stretching of the original length ##l_{0}##. In B we add a weight ##mg## which force is balanced by ##kl## (Hooke's Law; the...
  22. N

    Partial Differential Equation in Special Relativity

    Homework Statement (a) Light waves satisfy the wave equation ##u_{tt}-c^2u_{xx}## where ##c## is the speed of light. Consider change of coordinates $$x'=x-Vt$$ $$t'=t$$ where V is a constant. Use the chain rule to show that ##u_x=u_{x'}## and ##u_{tt}=-Vu_{x'}+u_{t'}## Find ##u_{xx},u_{tt},##...
  23. E

    Conserved quantities in the Korteweg-de Vries equation

    Homework Statement Consider the Kortweg-de Vires Equation in the form $$\frac{\partial \psi}{\partial t}+\frac{\partial^3 \psi}{\partial x^3}+6\psi\frac{\partial \psi}{\partial x}=0$$ Find the relation between the coefficients ##c## and ##d## , such that the following quantity is conserved...
  24. Michii

    I Differential equation of all the conics in the plane

    Hi, the problem is parametric families: To find Differential equation of all the conics in the plane with the origin in the center But when you speak of center at the origin being the equation of the conics: Ax ^ 2 + Bxy + cy ^ 2 + Dx + ey + F, is it correct to take the origin by making x and...
  25. D

    I Lambert's Law differential equation

    22. According to Lambert's law of absorption, the percentage of incident light absorbed by a thin layer of translucent material is proportional to the thickness of the layer. If sunlight falling vertically on ocean water is reduced to one-half its initial intensity at a depth of 10 feet, at...
  26. V

    I Classification of differential equation

    Hi, I have an equation that takes the form: ax''-by' + c = 0 where x'' is second order with respect to time and y' is first order with respect to time. Would this be classed as a partial differential equation? Thanks very much for your help :)
  27. Euler2718

    (Ordinary) Differential Equation Trouble

    Homework Statement Find the solution of the differential equation by using appropriate method: t^{2}y^{\prime} + 2ty - y^{3} = 0 Homework Equations I'm thinking substitution method of a Bernoulli equation: v = y^{1-n} The Attempt at a Solution [/B] t^{2}y^{\prime} + 2ty - y^{3} = 0...
  28. Drakkith

    Differential Equation Initial Value Problem

    Homework Statement A Solve the following initial value problem: ##\frac{dx}{dt}=-x(1-x)## ##x(0)=\frac{3}{2}## B. At what finite time does ##x→∞## Homework EquationsThe Attempt at a Solution ##\frac{dx}{dt}=x(x-1)## ##\frac{dx}{x(x-1)}=dt## Partial fractions...
  29. S

    I Solve Nonlinear DE: Friedmann Eqns for H 0-10^7

    From cosmology, the friedmann equations are given by, ##H^2 = (\frac{\dot a}{a})^2 = \frac{8\pi G}{3} \rho \, , \quad \frac{\ddot a}{a} = -\frac{4\pi G}{3}(\rho+3p) \, , \quad## where ##\rho = \frac{1}{2}(\dot \phi^2 + \phi^2)## and ##p = \frac{1}{2}(\dot \phi^2 - \phi^2)## To get ##\dot H##...
  30. F

    Differential equation of gradually varied flow

    Homework Statement I have no idea how the third formula of dy/dx is derived ... Homework EquationsThe Attempt at a Solution I know that the Q = (1/n)(A)(R^2/3) [(s)(^0.5)] , Q = K [(s)(^0.5)] , so , K= (1/n)(A)(R^2/3) i know that for very wide channel , y = R A = by K= (1/n)(A)(R^2/3)...
  31. cheapstrike

    Doubt related to formation of a differential equation

    Homework Statement Find the order of the differential equation of y=C1sin2x+C2cos2x+C3. Homework Equations - The Attempt at a Solution [/B] I read in my book that the order of the differential equation is equal to the number of arbitrary constants but the answer given is 2. Btw I have...
  32. R

    Differential Equation: x^2y''-xy'-3y=2x^-(3/2)

    Homework Statement Homework EquationsThe Attempt at a Solution I am not asking to find the answer, just wanted to know whether to use the variation of parameters or undetermined coefficients. Because this was on a test problem and I used variation of parameters instead. I know it is a...
  33. R

    Please help with this Differential equation problem

    Homework Statement y''-16y=2e^4x. Find general solution Homework EquationsThe Attempt at a Solution I have the homogenous equation which is c1e^-4x+c2e^4x, but I'm trying to find the particular solution for 2e^4x. I did yp=ae^4x, yp'=4ae^4x, yp''=16ae^4x, then plugged it into the equation...
  34. R

    Need help solving Differential Equation

    Sin(3x)dx+2ycos(3x)dy=0 So far, I have ∫2ydy=-∫sin(3x)/cos^2(3x)dx. Is that right? If so, how do I integrate sin(3x)/cos^2(3x)?
  35. T

    I About stochastic differential equation and probability density

    I have two questions about the use of stochastic differential equation and probability density function in physics, especially in statistical mechanics. a) I wonder if stochastic differential equation and PDF is an approximation to the actual random process or is it a law like Newton's second...
  36. J

    Change of variables in a differential equation

    Homework Statement Transform the equation: x2 * d2y/dx2 + 2 * x * dy/dx + (a2/x2)*y = 0 Using: x=1/t Homework Equations The differential of a function of several variables, and the common rules of differentiation. https://en.wikipedia.org/wiki/Derivative The Attempt at a Solution As...
  37. M

    B Why is the use of absolute value in vector norms a matter of preference?

    I would like to ask you why the author does not use absolute value of y instead of y? Source: Mathematical Methods in the Physical Sciences by Mary L. Boas Thank you.
  38. OneByBane

    A Solve DE: m, h, E, Z, t, k, a (1-4)

    I am currently trying to solve this differential equation: r2/F(r) d2F(r)/dr2 + 2mr2/h2(E + Zt2/kr) - a2 = 0 Wher m, h, E, Z, t and k are other variables and 'a' can have values 1, 2, 3, 4... (Whole numbers) I have come across this while solving a problem in physics and have no clue if this...
  39. Telemachus

    Solving partial differential equation numerically

    Hi there. I am trying to self teach how to solve partial differential equations numerically using finite differences. I know this is a complex field, that requires much more knowledge of the theory than what I actually know, but anyway I wanted to try. Anyway, I've tried to build my own...
  40. Kernul

    Hooke's Law Second Order Differential Equation

    Homework Statement A mass ##m## on a frictionless table is connected to a spring with spring constant ##k## so that the force on it is ##F_x = -kx## where ##x## is the distance of the mass from its equilibrium position. It is then pulled so that the spring is stretched by a distance ##x## from...
  41. S

    Series solution for differential equation

    <OP warned about not using the homework template> Obtain a series solution of the differential equation x(x − 1)y" + [5x − 1]y' + 4y = 0Do I start by solving it normally then getting a series for the solution or assume y=power series differentiate then add up the series? I did the latter and...
  42. R

    I 2nd order differential equation problem with sin(theta)

    I have a differential equation that is essentially this: θ''(t)=c*sin[θ(t)] . I've been stymied trying to find a solution, and even when I tried using Maple, I got a nasty integral of a Jacobian amplitude. I'm tempted to use a small angle approximation, but the angle is 0≤θ≤π/2. I know this...
  43. Mr Davis 97

    Construct a differential equation from the basis of solution

    Homework Statement Write down a 3x3 matrix A such that the equation ##\vec{y}'(t) = A \vec{y}(t)## has a basis of solutions ##y_1=(e^{-t},0,0),~~y_2 = (0,e^{2t},e^{2t}),~~y_3 = (0,1,-1)## Homework EquationsThe Attempt at a Solution I was thinking that, it looks like the matrix would have to...
  44. Cocoleia

    Non exact differential equation, initial value problem

    Homework Statement I am trying to solve the following: y'''-9y'=54x-9-20e^2x with y(0)=8, y'(0)=5, y''(0)=38 Homework EquationsThe Attempt at a Solution The right answer is: y= 2+2e^3x+2e^(-3x)-3x^2+x+2e^2x I am only wrong on the coefficients C2 and C3. Where did I mess up in my solution?
  45. S

    Unstable solution of differential equation

    Homework Statement Consider the following differential equation $$\frac{\partial^{2}\phi}{\partial t^{2}}-\nabla^{2}\phi = \phi(a-b\phi^{2}), \qquad a>0.$$ I would like to prove that $\phi=0$ is an unstable solution of this equation. Homework Equations The Attempt at a Solution Do you...
  46. Cocoleia

    Initial value problem - differential equations

    Homework Statement I am given (y^2 + y sin x cos y) dx + (xy + y cos x sin y) dy = 0, y(0) = π/2 . I need to solve this Homework EquationsThe Attempt at a Solution At this point they still aren't exact, so I gave up. I can't figure out what the problem is. Is it possible that I have to...
  47. G

    A Flatness Problem Differential Equation

    Hello to everybody, Since 3 days, I've been trying to obtain Eq. (173) from https://ned.ipac.caltech.edu/level5/Sept03/Trodden/Trodden5.html I know I need to derivate with respect to a, but it is being impossible to obtain the final answer. Can anybody help me with a detailed derivation...
  48. G

    Resonance in forced oscillations

    Homework Statement Consider the differential equation: mx'' + cx' + kx = F(t) Assume that F(t) = F_0 cos(ωt). Find the possible choices of m, c, k, F_0, ω so that resonance is possible. Homework EquationsThe Attempt at a Solution I know how to deal with such problem when there is no damping...
  49. john augustine

    Dy/dx = xe^(y-2x), form differntial eqaution

    dy/dx = xe^(y-2x) , i am asked to form differential equation using this equation . the ans given is (e^-y) = 0.5(e^-2x)(x+0.5) + a , how to get the answer? btw , i have attached my working
  50. W

    MHB Solve Diff. Eq. for g(x) w.r.t f(x) & c

    In the following equation: $$g'(x) = f'\left( x + \frac{c f'(x)}{\sqrt{ 1 + f'(x)^2 }} \right)$$ find $g(x)$ with respect to $f(x)$ where $c$ is any constant.
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