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. WMDhamnekar

    MHB The Jacobian and area differential

    I don't understand the following definition. If we let $u=\langle u,v \rangle$ , $p=\langle p,q\rangle,$ $x=\langle x,y \rangle$,then (x,y)=T(u,v) is given in vector notation by x=T(u). A coordinate transformation T(u) is differentiable at a point p , if there exists a matrix J(p) for which...
  2. barryj

    A Hint as to how to solve an integral differential equation

    I know how to solve a differential equation using Eulers method but what if the equation has an integral part? i.e. a RLC electrical circuit. Vsource = iR + L di/dt + (1/C)int i dt can this be done? a link to how to solve this would be helpful.
  3. E

    I How do I find the integrating factor for a differential equation?

    So I've been Stuck On this Equation Trying to find the integrating factor (im not sure if it has one) appreciate the help
  4. L

    B Line Integral, Dot Product Confusion

    From my interpretation of this problem (image attached), the force applied to the point charge is equal and opposite to the repulsive Coulomb force that that point charge is experiencing due to the presence of the other point charge so that the point charge may be moved at a constant velocity. I...
  5. Nick Amos

    Tabular Method for a Unique Spur Gear Differential System

    Hello everyone, this is my first post here so please excuse me if its in the wrong area. I am looking for the tabular approach to solve a complex gear train based off the spur gear differential design. Attached is a photo of the differential. The requirements of the necessary differential...
  6. WMDhamnekar

    MHB Difficult first order linear differential equation

    Hello, I want to solve the following differential equation. $y'=\dfrac{x^3-y^3}{x-y}$. How to solve it?
  7. opus

    Use of the constant C in the solution of Differential Equations

    Homework Statement I put this is the Calculus section because it relates to Calculus I and if I put it in Diff Eq section I think it would be assumed that I know the necessary terms, etc. My question is in regards to the use of the constant ##C## in differential equations. For reference, the...
  8. T

    I Solution to a second order differential equation

    I have currently been reading a book called 'Mathematical Methods In Physical Sciences'. Whilest I was looking at the differential section I came across a differential which I have never thought about before, which is of the form...
  9. K

    I Understanding Differential Forms and Basis Vectors in Curved Space

    In the exercises on differential forms I often find expressions such as $$ \omega = 3xz\;dx - 7y^2z\;dy + 2x^2y\;dz $$ but this is only correct if we're in "flat" space, right? In general, a differential ##1##-form associates a covector with each point of ##M##. If we use some coordinates...
  10. V

    Geometry Classical and modern differential geometry

    Im planning on taking a course on classical differential geometry next term. This is the outline: The differential geometry of curves and surfaces in three-dimensional Euclidean space. Mean curvature and Gaussian curvature. Geodesics. Gauss's Theorema Egregium. The textbook is "differential...
  11. H

    Partial Differential Equation Mathematical Modelling

    Salutations, I have been trying to approach a modelling case about organism propagation which reproducing with velocity $$\alpha$$ spreading randomly according these equations: $$\frac{du(x,t)}{dt}=k\frac{d^2u}{dx^2} +\alpha u(x,t)\\\ \\ u(x,0)=\delta(x)\\\ \lim\limits_{x \to \pm\infty}...
  12. Edge5

    I Solution of Quantum differential equation

    (I think I couldn't add the image) you can see my answer in link https://pasteboard.co/HPKZ6KD.jpg (Please first see my answer in the link) But in answer it is φ= Asin(kx) + Bcos(kx) I know that euler formula is eix = cosx +isinx But I can't get this answer can you help me?
  13. E

    A Differential Equation to Difference Equation

    Hi all, I am a bit new in this, am trying to learn DE, dynamical systems, & chaos. I am looking into some answers for the following questions: 1) Is it always possible to derive a difference equation for every differential equation, and if so how do we do that? 2) Consider Lorenz system...
  14. F

    Solve the second-order differential equation

    Homework Statement Homework Equations The Attempt at a Solution Can someone check my answer please ?
  15. F

    Solve the differential equation with constant coefficients

    Homework Statement Homework Equations The Attempt at a Solution [/B] Is my answer correct?
  16. M

    Solving Differential Equation with Frequency Response

    Homework Statement The AC response of an inductor can be modeled by the following differential equation: L \frac{di}{dt} + iR = V Find, using frequency response, the current of the system when the applied voltage V is: V = V_0 \sin(\omega t) Homework EquationsThe Attempt at a Solution In...
  17. Chromatic_Universe

    I Solving a nonlinear first order differential equation

    (a'[t]/a[t])^2 == K*(A + B*a[t]^-6)^1/2} is the equation to be solved for getting the solution of a(t) in terms of time(t). Any ideas on how to solve this problem? Use of Matlab or Mathematica is accepted.
  18. WMDhamnekar

    MHB Solutions of DE System & 2nd Order Differential Equation

    Hello, $\vec{x'}=\small\begin{pmatrix}1&2\\3&2\end{pmatrix}\vec{x}+t\small\begin{pmatrix}2\\-4\end{pmatrix}$ Now i got the solution to this differential equation system as...
  19. S

    Differential equation with power series method

    Homework Statement I need to solve the DE y’ = x^2y using the power series method Homework Equations y = sum(0->inf)Cnx^n y’ = sum(1->inf)nCnx^(n-1) The Attempt at a Solution I plug in the previous two equations into the DE. What is the general procedure for these problems after that...
  20. S

    Prove Sturm-Liouville differential operator is self adjoint.

    Homework Statement Prove Sturm-Liouville differential operator is self adjoint when subjected to Dirichlet, Neumann, or mixed boundary conditions. Homework Equations l = -(d/dx)[p(x)(d/dx)] + q(x) The Attempt at a Solution I have no idea. If someone can give me a place to start that would...
  21. Hawkingo

    Help in solving an inexact differential equation

    Homework Statement The question is to solve the inexact equation by turning it into exact.the equation is ##( x + y + 4 ) d x + ( - x + y + 6 ) d y = 0## Where "x" and "y" are variable. 2. Homework Equations [/B] 1.(x+y+4)=m and (-x+y+6)=n 2.Integrating Factor =##\frac { 1 } { x ^ { 2 } + y...
  22. Kaguro

    A differential equation with undetermined coefficients

    Homework Statement Solve the following DE with the method of undetermined coefficients: y'' + 4y = 2cos(3x)cos(x) Homework Equations 2cos(3x)cos(x) = cos(4x) + cos(2x) The Attempt at a Solution Let's split the particular integral into two parts: yp1 and yp2. So yp1 is solution for RHS=cos(4x)...
  23. Biker

    I Using Differential operators to solve Diff equations

    I don't really understand how their inverses work. For example, in solving 2nd order linear non-homogeneous differential equations. The particular solution is found by ## y_{pi} = \frac{p(x)}{f(D)} ## And they continue by expanding using maclaurin series. How do you treat an operator as a...
  24. C

    Transfer Function of a FET Differential Amplifier

    Homework Statement [Taken from Razavi's Design of Analog CMOS Integrated Circuits 2nd edition]Homework EquationsThe Attempt at a Solution [/B] I'm not too sure what to do with this question. Here's what I think of the circuit This doesn't look like a differential amplifier to me since...
  25. Felipe Lincoln

    First order differential equation

    Homework Statement Solve the following differential equation such that ##x(0)=1##. ## \dfrac{dx}{dt} + 2tx = 3e^{-t^2}+t## Homework Equations Integrating factor: ##\mu(t) = exp\left(\int_0^t2t \right)## The Attempt at a Solution I used the integrating factor and then got the solution ##x(t) =...
  26. N

    1st Order Differential Equation - Power Series Method

    Homework Statement The Attempt at a Solution I have deliberately made several obvious steps, because I keep ending up here. However I have no idea what to do from here. I thought about the fact, that differential equations have the solution ##x = x_{HOM} + x_{Inhom}##, but the ##x_{HOM}##...
  27. A

    MHB Partial differential equations problem - finding the general solution

    4\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x} = 3u , u(x,0)=4e^{-x}-e^{-5x} let U =X(x)T(t) so 4X\frac{\partial T}{\partial t}+T\frac{\partial X}{\partial x} = 3XT 4\frac{\partial T}{T \partial t}+\frac{\partial X}{X \partial x} = 3 \left( 4\frac{\partial T}{T...
  28. Josu Aguirrebeitia

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    Hi. After arranging the dynamic contact between a elastic ball against a flat, I have reached the following differential equation for the motion during the contact: m·x’’+(k+c·x’)·x^n=0 with m,c,k>0 and for exponent n --> 1<n<2 Any functional form for this equation? I have solved it...
  29. K

    Geometry Vargas' book about Differential Geometry

    I'm learning Differential Geometry (DG) on my own (I need it for robotics). I realized that there are many approaches to DG and one is Cartan's, which is presented in Vargas's book. I think that book is highly opinionated, but I don't know if that's a good or bad thing. Does anyone of you know...
  30. cianfa72

    I Differential structure on a half-cone

    Hi, consider an "half-cone" represented in Euclidean space ##R^3## in cartesian coordinates ##(x,y,z)## by: $$(x,y,\sqrt {x^2+y^2})$$ It does exist an homeomorphism with ##R^2## through, for instance, the projection ##p## of the half-cone on the ##R^2## plane. You can use ##p^{-1}## to get a...
  31. G

    Transmission line: leakage current differential equation

    Homework Statement I have a coaxial cable with internal conductor of radius r1 and external conductor of radii r2 and r3. The material of the conductors has a conductivity ##\sigma_1##. Between the conductors there is a imperfect dielectric of conductivity ##\sigma_2##. Consider the...
  32. M

    A Differential of a function

    We define the differential of a function f in $$p \in M$$, where M is a submanifold as follows In this case we have a smooth curve ans and interval I $$\alpha: I \rightarrow M;\\ \alpha(0)= p \wedge \alpha'(0)=v$$. How can I get that derivative at the end by using the definitions of the...
  33. M

    A Differentiability of a function between manifolds

    Hello, let $$M^n \subset \mathbb{R}^N$$ $$N^k \subset \mathbb{R}^K$$ be two submanifolds. We say a function $$f : M \rightarrow N$$ is differentiable if and only if for every map $$(U,\varphi)$$ of M the transformation $$f \circ \varphi^{-1}: \varphi(U) \subset \mathbb{R}^N \rightarrow...
  34. R

    I Differential of the coordinate functions

    Hello folks, I'm glad that I discovered this forum. :) You might save me. I'm hearing right now differential geometry and am having some problems with the subject. May you explain me the follwoing. We had the special case of the i-th projection. My lecturer now posited that the differential of...
  35. K

    Differential equation for the acceleration of an oscillating particle

    Homework Statement acceleration of certain oscillating particle described by a = -x/9 determine the position of this particle when t = 3π/2 if when t=0 x=0 and v=v0 Homework Equations dv/dt=a The Attempt at a Solution frankly I am not sure how to start but i have two ways in my mind(even i...
  36. H

    MHB Help: Differential equation Romeo & Juliet

    Hi, I have to make an assignment on differential equations and Romeo and Juliet. r(t) is romeo's love for Juliet at time t, j(t) is Juliet's love for Romeo at time t So far, it is given: dr/dt=-j and dj/dt=r. It is also given that Romeo & Juliet's families are enemies, thus the initial...
  37. M

    MHB Differential equations Romeo and Juliet

    Hi all! I need to give a presentation about a problem in class, but I can't seem to figure it out. This is the problem: Consider the system dr/dt = -j, dj/dt = r , where r (t) represents Romeo’s love (positive values) or hate (negative values) for Juliet at time t, and j(t) similarly...
  38. T

    Find v(t) from Newton's Second Law and Differential Equation

    <Moderator's note: Moved from a technical forum and thus no template.> Is what I have done correct ? I want to find v(t) from Sigma F = m*a. I have gravity force mg pointing downward with positive direction and resistive force R = -b*v^2 pointing upwards with negative direction are acting on a...
  39. beefbrisket

    I Sign mistake when computing integral with differential forms

    The question provides the vector field (xy, 2yz, 3zx) and asks me to confirm Stokes' theorem (the vector calc version) but I am trying to use the generalized differential forms version. So, I am trying to integrate \omega = xy\,dx + 2yz\,dy + 3zx\,dz along the following triangular boundary...
  40. K

    Understanding Zero Value Differential in Euler Number Equations

    Homework Statement If a = 9-v² then prove that v = 3 (e^6t - 1)/(e^6t + 1) the condition when t=0 also v has zero value Homework Equations I don't quite understand in this but general equation should be dv/dt = a The Attempt at a Solution Actually i don't don't have any idea in this problem...
  41. Gene Naden

    I Differential for surface of revolution

    O'Neill's Elementary Differential Geometry contains an argument for the following proposition: "Let C be a curve in a plane P and let A be a line that does not meet C. When this *profile curve* C is revolved around the axis A, it sweeps out a surface of revolution M." For simplicity, he...
  42. RpWinter

    Solving the differential equation of planetary motion

    Hey, this is how i tried solving the differential equation The solution however does not match the general solution of the equation. Also differentiating it twice does not give me the previous equation. Please tell me if i did some mistake while solving. I already know how to solve by finding...
  43. Avatrin

    Motivating definitions from differential geometry

    Hi I have always had an issue with understanding the definitions used in mathematics. I need examples before I can start using and reasoning with them. However, with tensor products, I have been completely stuck. Stillwell's Elements of Algebra was that made abstract algebra "click" for me...
  44. C

    I Christoffel symbols knowing Line Element (check my result)

    Hi! I'm asked to find all the non-zero Christoffel symbols given the following line element: ds^2=2x^2dx^2+y^4dy^2+2xy^2dxdy The result I have obtained is that the only non-zero component of the Christoffel symbols is: \Gamma^x_{xx}=\frac{1}{x} Is this correct? MY PROCEDURE HAS BEEN: the...
  45. Bill2500

    I Topology vs Differential Geometry

    Hello. I am studying Analysis on Manifolds by Munkres. My aim is to be able to study by myself Spivak's Differential Geometry books. The problems is that the proof in Analysis on Manifolds seem many times difficult to understand and I am having SERIOUS trouble picturing myself coming up with...
  46. G

    Calculus Ordinary and partial differential equations

    Hi, I'm attempting to learn differential equations on my own. Does anyone recommended a textbook that comes with/has a solution manual? I learn faster when I can see a problem worked out if I can't solve it. Thanks.
  47. komarxian

    Differential Equations: Solve the following

    Homework Statement Solve the following differential equations/initial value problems: (cosx) y' + (sinx) y = sin2x Homework Equations I've been attempting to use the trig ID sin2x = 2sinxcosx. I am also trying to solve this problem by using p(x)/P(x) and Q(x) The Attempt at a Solution...
  48. komarxian

    Differential Equations, solve the following: y^(4) - y'' - 2y' +2y = 0

    Homework Statement Solve the following differential equations/initial value problem: y^(4) - y'' - 2y' +2y = 0 Hint: e^-x sinx is a solution Homework Equations I was attempting to solve this problem by using a characteristic equation. The Attempt at a Solution y'''' -y'' -2y' + 2y = 0 -->...
  49. R

    MHB Differential equation with a matrix

    Suppose we have the matrix $ \mathbf{N} = \begin{bmatrix} 4 & -2 \\ -2 & 1 \end{bmatrix}$ and $\mathbf{X} = \begin{bmatrix}x \\ y \end{bmatrix}$. I want to solve $\displaystyle \frac{d\mathbf{X}}{dt} = \mathbf{NX}$. The eigenvalues of the matrix are $\lambda_1, \lambda_2 = 0,5$ and eigenvectors...
  50. D

    I Solution:Second Order Linear Non-Homogenous ODEs in Physics

    Hello, could someone please give me some examples of where order linear non homogenous ordinary differential equations are used in physics[emoji4]
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