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. S

    A Differential forms and vector calculus

    Let ##0##-form ##f =## function ##f## ##1##-form ##\alpha^{1} =## covariant expression for a vector ##\bf{A}## Then consider the following dictionary of symbolic identifications of expressions expressed in the language of differential forms on a manifold and expressions expressed in the...
  2. S

    A Line integrals of differential forms

    Consider a curve ##C:{\bf{x}}={\bf{F}}(t)##, for ##a\leq t \leq b##, in ##\mathbb{R}^{3}## (with ##x## any coordinates). oriented so that ##\displaystyle{\frac{d}{dt}}## defines the positive orientation in ##U=\mathbb{R}^{1}##. If ##\alpha^{1}=a_{1}dx^{1}+a_{2}dx^{2}+a_{3}dx^{3}## is a...
  3. 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?
  4. DarkBabylon

    Solving differential equations using numeric methods

    Hello, I have been working on a little movement system in a program called Game Maker: studio. The code works fine on the programming perspective, but something I did not expect happened: When I ran the code by adding to the speed while pressing a key, and every step passively subtracting from...
  5. J

    I Help to understand second order differential definition

    Hi folks, I am reading Poisson's Teatrise on Mechanics. In the introduction he talks about the infinitesimals. Let's say A is a first order infinitely small quantity, a differential of the first order, if the ratio of A to B is infinitely small too it means B is an infinitesimal of the second...
  6. W

    Linear Differential Equations and Linear Operator Problem

    Homework Statement I'm not sure how to approach this. The question involves linear operators and a non-homogenous differential equation. Here is the question: https://s15.postimg.org/cdmw80157/Capture.png Homework Equations They are given in the question The Attempt at a Solution I really...
  7. K

    I Differential Forms: Definition & Antisymmetric Tensor

    Why does the definition of a differential form requires a totally antisymmetric tensor?
  8. Kanashii

    Solve for the solution of the differential equation

    Homework Statement Solve for the solution of the differential equation and use the method of variation of parameters. x`` - x = (e^t) + t Homework Equations [/B] W= (y2`y1)-(y2y1`) v1 = integral of ( g(t) (y1) ) / W v2 = integral of ( g(t) (y2) ) / W The Attempt at a Solution [/B] yc= c1...
  9. S

    Find the exact value of a differential equation.

    Homework Statement dy/dx= 200-2y. y(0)=75 Homework EquationsThe Attempt at a Solution Do you move dx over and integrate. Do you just integrate it 200y-y^2+c
  10. G

    I Differential form of Gauss' law: All three terms the same value?

    Hi. Is the Maxwell equation $$\nabla\cdot\vec{E}=\frac{\rho}{\varepsilon_0}$$ even true in the stronger form $$\frac{\partial E_i}{\partial x_i}=\frac{\rho}{3\cdot\varepsilon_0}\enspace ?$$ I guess not, since I haven't found a source suggesting this. But shouldn't the isotropic electric field...
  11. arpon

    Prove dQ is an inexact differential

    Homework Statement ##dz=Mdx+Ndy## is an exact differential if ##(\frac{\partial M}{\partial y})_x=(\frac{\partial N}{\partial x})_y##. By invoking the condition for an exact differential, demonstrate that the reversible heat ##Q_R## is not a thermodynamic property. Homework Equations...
  12. 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}...
  13. B

    Studying How to Learn both Differential Geometry and Relativity?

    Dear Physics Forum personnel, Is it possible to learn differential geometry simultaneously while learning the relativity and gravitation? I have been reading Weinberg's book (currently in Chapter 02), but I believe that modern research in relativity is heavily based on the differential...
  14. K

    A Need help with a differential equation

    Hi all, I have derived a differential equation, which I don't know how to solve. I can do some numerical simulations, but would really be interested in, at least, knowing if an analytical solution exists, so would appreciate any help with it: (I have removed argument from y)...
  15. 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...
  16. Dusty912

    Solving Bernoulli's Differential Equation

    Homework Statement xy(dx)=(y2+x)dyHomework Equations integrating factor : u(x)=e∫p(x)dx standard form of linear DE: dy/dx + P(x)y=Q(x) standard form of bernoulli's differential equation: dy/dx + P(x)y=Q(x)yn change of variables v=y1-n The Attempt at a Solution xy(dy)=(y2+x)dx xy(dy/dx)=y2 +x...
  17. K

    B Private solution to a polynomial differential equation

    The polynomial equation and it's private solution: $$(1)~~ay''+by'+cy=f(x)=kx^n,~~y=A_0x^n+A_1x^{n-1}+...+A$$ If i, for example, take ##f(x)=kx^3## i get, after substituting into (1), an expression like ##Ax^3+Bx^2+Cx+D## , but that doesn't equal ##kx^3##
  18. 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
  19. Dusty912

    Separable Differential Equation

    Homework Statement Solve the differential equation: (ex+1)cosy dy + ex(siny +1)dx=0 y(0)=3 Homework Equations none The Attempt at a Solution (ex+1)cosy dy + ex(siny +1)dx=0 (ex+1)cosy dy =- ex(siny +1)dx cosy/(siny+1)dy=-ex/(ex+1)dx ∫cosy/(siny+1)dy=-∫ex/(ex+1)dx using u sub on both the...
  20. Dusty912

    Separable differential equation

    Homework Statement Solve each of the following differential equations: 4xydx + (x2 +1)dy=0Homework Equations None The Attempt at a Solution 4xydx + (x2 +1)dy=0 (x2 +1)dy=-4xydx dy/y=-(4xdx)/(x2 +1) ∫dy/y=∫-(4xdx)/(x2 +1) ln|y|=-2ln|x2+1| +C used u-sub on last step fo u=x2 +1
  21. O

    Solve the differential equation F=F0+kv

    Homework Statement Find the velocity of v as a function of displacement x for a particle of mass m which starts from rest at x=0 and subject to the following force: F=F_0+kv You could say mv = F0*t + kx, but the answer in the back of the book is an equation that is only in terms of x and v...
  22. Pouyan

    Differential equations and geometric series

    Homework Statement I Have a differential equation y'' -xy'-y=0 and I must solve it by means of a power series and find the general term. I actually solved the most of it but I have problem to decide it in term of a ∑ notation! Homework Equations y'' -xy'-y=0 The Attempt at a Solution I know...
  23. Phylosopher

    Courses Differential equations theory course, is it useful?

    Hello,I have a department elective course called "Differential equations theory" but I have no idea if it is going to be useful for me as a physicist (I'm interested in the theory/ minor math). The description of the course is as follows : The fundamental theorem of existence and autism, linear...
  24. A

    Linear differential equation hwk check

    1. Homework Statement Posted Homework Equations Posted The Attempt at a Solution Posted I just need the work to be checked.
  25. D

    I Lie derivative of a differential form

    Hello, I have a maybe unusual question. In a paper, I recently found the equation $$\mathcal{L}_v(v_i dx^i) = (v^j \partial_j v_i + v_j \partial_i v^j) dx^i$$ Where v denotes velocity, x spatial coordinates and \mathcal{L}_v the Lie derivative with respect to v. Now I'm an undergraduate who...
  26. D

    Separating vector differential equation into components

    Homework Statement Write down the component form of the differential equations of motion of a projectile if the air resistance is proportional to the square of the speed. Are the equations seperated? Show that the x component of the velocity is given by \dot{x}=\dot{x}_0e^{^-\gamma s} where s...
  27. M

    How do I enter a differential equation in maple?

    Hello, I am currently taking ODE's and the class has an optional lab to accompany it. So far in the lab we've been doing some pretty basic stuff. But we've finally moved on to entering in differential equations, and I'm confused. 1. Homework Statement dydx+2x=2y How do I enter this equation...
  28. J

    A Second Order Differential equation Bessel-type

    Hello! Im trying to solve this second order differential equation: \begin{equation*} -\dfrac{d^2y}{dx^2}+\dfrac{3}{x}\dfrac{dy}{dx}+(x^2+gx^4+2)y=0 \end{equation*} Any idea? Maybe it could be converted to a Bessel-like equation (?) with an appropriate change of variables. The equation...
  29. nysnacc

    Differential equation uniqueness

    Homework Statement Homework Equations Leibniz notation: dy/dx = f(x) g(y) integral 1/g(y) dy = integral f(x) dx The Attempt at a Solution integral 1/y dy = integral sqrt (abs x) dx ln (y) = ? because sqrt (abs x) is not integrable at x =0 Then my thought is that y=0 is not unique
  30. Mr Davis 97

    Differential equation for air resistance

    Homework Statement Solve the differential equation ##\displaystyle Cv^2 - mg = m\frac{d^2 y}{dt^2}## Homework EquationsThe Attempt at a Solution The problem is nonlinear, so we need to use unconventional methods. Specifically, if we can express the derivative of y with respect to v, then we...
  31. T

    Finding a Solution to a System of Differential Equations

    Homework Statement Find a solution \bf{\phi} of the system $$y'_1(t)=y_1(t)+y_2(t)+f(t)$$ $$y'_2(t)=y_1(t)+y_2(t)$$ where f(t) is a continuous function $$\bf{\phi} (0)=(0,0)$$ Homework Equations A hint was given to define ##v(t)=y_1(t)+y_2(t)## The Attempt at a Solution Using the suggested...
  32. B

    Geometry Regarding to Spivak's Differential Geometry trilogy

    I would like to begin my first exploration of the arts of differential geometry/topology with the first volume of M. Spivak's five-volume set in the different geometry. Is a thorough understanding of vector calculus must before reading his book? I read neither of his Calculus nor Calculus on...
  33. R

    MHB Differential Equation: Tangents & Normals to y=x^2

    Problem:Find the differential equation satisfied (i) by the equation of the family of tangents to y=x^2 and (ii) by the equation of the family of normals to y=x^2.
  34. B

    Confused on simple differential equation

    Homework Statement Find the values of m so that ##y = x^m## is a solution of ##x^2\frac{d^2y}{dx^2} - 3x\frac{dy}{dx} -12y = 0## Homework Equations ##y = x^m## ##y'=mx^{m-1}## ##y''=(m^2-m)x^{m-2}## The Attempt at a Solution So after plugging and chugging we get $$(m+2)(m-6)x^m = 0 $$...
  35. jdawg

    Memorizing solutions for differential equations

    Homework Statement [/B] Its been a pretty long time since I've taken differential equations and I'm expected to know the solutions to the kinds of DEs below for my fluid mechanics class. In class my professor worked a 2nd order DE: dy2/dx2 = -k2*y and told us the way to think about it was to...
  36. cosmonium

    I Can Heat Differential Generate Thrust?

    This is an experiment I have wanted to do for a few years now but don't have the necessary equipment. GR tells us if you have identical objects with the same weight exactly when they are at the same temperature, then when one object is heated, it will weigh more. This is because the...
  37. tomwilliam2

    I Solving linear differential equations

    I have the following linear differential equations: ##A\dot{x} + By = 0## ##C\dot{y} + Dx = 0## Where x and y are functions of t, and A through D are constants. I can solve this fairly easily by differentiating the first equation, rearranging, and removing one of the variables, which gives me a...
  38. 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##...
  39. J

    Differential equation: solving for the General Solution

    Homework Statement [/B] (dy/dx)^2 = (1-y^2) / (1-x^2)Homework Equations Separating the variables I arrive with: dy/sqrt(1-y^2) = dx/sqrt(1-x^2) By integration on both sides by trigonometric substitution and putting it in a general solution: Arcsin y - Arcsin x = C The Attempt at a Solution If...
  40. E

    I I with differential equation(3x^2-2y^2)(dy/dx)=2xy

    I have been stuck in this problem for two days now. I am starting DE this semester and I want to move ahead. so for this problem I attempted to use x=yv, then v=x/y. so I move dy/dx to the other side of the equation and divide by 2xy both sides, which leads to (3x^2-2y^2)/(2xy)=dx/dy. then...
  41. H

    I Multi variable differential

    Hi I understood above differential ## typo. RHS= 2*f ' (x,2z-x) but, what is answer of below equation? is this right? f ' ( g(z),2z-x) * g' (z) + f ' ( g(z),2z-x) *2
  42. C

    Linear application and différential

    Homework Statement Let be E a normded vectoriel space, ##dim(E) = m \in \mathbb{N}^{*}##, I have to show that ##\exists \rho_{1}, \rho_{2} > 0 | \forall u \in L(E), ||u^{m} - Id| \leq \rho_{1} \Rightarrow |u - Id| \leq \rho_{2}##. Homework Equations Nothing. The Attempt at a Solution [/B] I...
  43. 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...
  44. J

    Matrix riccati differential equation using matlab

    Homework Statement Homework EquationsThe Attempt at a Solution
  45. F

    Ambiguity in the method applied for differential equations

    Homework Statement Why do we need two solutions to solve a 2nd order linear differential equation? lets consider a differential equation with equal roots for auxiliary equation. So the reasoning behind why can't we use y=Aen1x+Ben2x as its general solution is because since the roots are equal...
  46. 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...
  47. P

    B Exploring the Meaning of "d" in Differential Physical Quantities

    [Mentor's note: This question was split off from: https://www.physicsforums.com/threads/meaning-of-physical-quantities-and-division.880214/] What does this differential "d" mean? Why say dE instead of E? or
  48. Jianphys17

    Differential Geometry book with tensor calculus

    Hi, there is a book of dg of surfaces that is also about tensor calculus ? Currently i study with Do Carmo, but i am looking for a text that there is also the tensor calculus! Thank you in advance
  49. 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!
  50. awholenumber

    Calculus First order differential equations ?

    what is a good book to learn first order differential equations ??
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