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. Q

    Help to reduce solution of differential equation: dy/dx=(xy+y)/(x+xy)

    ln(y) +y = ln(x) + x +C y=?
  2. Physics345

    Question regarding the differential height of mercury in a manometer

    Given: ##y_{2} - y_{1}= 0.2m## ##Q= 1.5\frac{m^{3}}{min}## ##Q= 0.025\frac{m^{3}}{s}## After conversion ##D_{1}= 0.2m## After conversion ##D_{2}= 0.1m## After conversion ##r_{1}= 0.1m## ##r_{2}= 0.05m## ##p_{1} - p_{2} = \frac{1}{2}P(v_{2}^{2}-v_{1}^{2}) + Pg(y_{2}-y_{1})## Calculating...
  3. Physics345

    Differential Equation ODE Solution help.

    dM/dY = x+2y+1 dN/dx = 1 (My-Nx)/n = 1 Integrating Factor => e^∫1dx= e^x (xye^x+ye^x+ye^x)dx + (xe^x+2ye^x)dy = 0 dM/dY =xye^x+e^x+2ye^x dN/dx = xye^x+e^x+2ye^x Exact ∫dF/dy * dy = ∫ (xe^x+2ye^x)dy F = xy*e^x + y^2*e^x + c(x) dF/dx = xy*e^x + y*e^x + y^2 * e^x + c'(x)...
  4. E

    MHB Total differential to calculate approximately the largest error

    I have the following problem: Use the total differential to calculate approximately the largest error at determine the area of a triangle rectangle (right triangle) from the lengths of the cathetus if they measure 6 and 8 cm respectively, with a possible error of 0.1 cm for each measurement. I...
  5. G

    I Find the general solution for the differential equation

    So in my previous math class I spotted on my book an exercise that I couldn't solve. We had to find the general solution for the differential equation. This was the exercise: 4y'' - 4y' + y = ex/2√(1-x2) Can anyone tell me how to solve this step by step?
  6. M

    I Projective Methods For Stiff Differential Equations

    Good evening, https://pdfs.semanticscholar.org/688b/e703a59a4a0c6fc96b4e42c38c321cd4d5b8.pdf Do you know :PROJECTIVE METHODS FOR STIFF DIFFERENTIAL EQUATIONS I have to make a program to solve a first-order differential equation according to this method but I do not arrive despite my efforts...
  7. V

    A Derivation of the differential Chapman-Kolmogorov Equation

    The integral equation is T(x_3,t_3|x_1,t_1)=\int \text{d}x_2T(x_3,t_3|x_2,t_2)T(x_2,t_2|x_1,t_1) where T(x_3,t_3|x_1,t_1) is the probability density of a Markov process taking the value x_3 at time t_3 given that it took the value of x_1 at time t_1. So far so good. To derive the differential...
  8. K

    Which Form of Maxwell's Equations is More Useful? (Integral versus Differential)

    There are two forms of the Maxwell equations, one is the differential form, the other is the integral form. Which one is more useful?
  9. B

    Linearizing the Lugiato-Lefever Partial Differential Equation

    I started by substituting the following anzatz: $$ \psi = \psi_e + \psi_1 $$ When ## |\psi_1| \ll 1 ##. Substituting the above into the equation yields: $$ \frac {d\psi_1} {dt} = -(1 + i\alpha)\psi_1 + \frac i 2 \frac {\partial ^ 2 \psi _1 } {\partial x ^ 2 } + i (\bar \psi_1 \psi_1 ^2 + \bar...
  10. D

    Need help with Matlab Function of Differential Equations

    WHAT HAPPENS IS That I need to model the example of A Protein G example, using a function f in Matlab, but when I execute the script, the graphics I get do not correspond to those of the example. The problem is that I can not understand what the model seeks to represent, besides that I do not...
  11. Protea Grandiceps

    I X variable in damping force equation for damped oscillation?

    Hi, for ease of reference this posting is segmented into : 1. Background 2. Focus 3. Question 1. Background: Regarding (one, linear, second-order, homogeneous, ordinary, differential) equation describing the force in a non-driven, damped oscillation: F = m.a = -k.x - b.v F =...
  12. A

    Trying to calculate the time derivative of a position differential

    here I am trying to find ##\frac{d}{dt}dx## where ##x(t)## is the position vector Now ##\frac{d}{dt}(v_x(x,y,z,t)dt)=\frac{dv_x}{dt}dt=\frac{\partial v_x}{\partial t}dt+\frac{\partial v_x}{\partial x}dx+\frac{\partial v_x}{\partial y}dy+\frac{\partial v_x}{\partial z}dz## Now dividing by ##dx##...
  13. M

    MHB Projective Methods for Stiff Differential Equations

    hello, now I'm working on a numerical method called: Projective Methods for Stiff Differential Equations: Problems with Gaps in Their Eigenvalue Spectrum. but I can not understand despite the hours of work that I spent on it I turn to you for help, applying this method on this exmple : y '= y...
  14. christang_1023

    Solve the differential equation of motional emf

    . Above is the figure of the problem. I am trying to solve x(t) and differentiate it to obtain v(t); however, I have difficulty solving the differential equation shown below. $$ v(t)=\int a(t)dt=\int \frac{B(\varepsilon-Blv)d}{Rm}dt \Rightarrow \frac{dx}{dt}=\frac{B\varepsilon...
  15. Clara Chung

    Question about differential cross section

    I have attached the two pages in my notes and I have the following question. 1. Where have the n_t*l gone in 9.9? (According to 9.5 why do they disappear?) 2. Why J_s=sigma_tot J_i? The dimension of flux is per m^2 and sigma is per area too, the dimension is not right...
  16. F

    Differential equation modeling glucose in a patient's body

    The rate at which glucose enters the bloodstream is ##r## units per minute so: ## \frac{dI}{dt} = r ## The rate at which it leaves the body is: ##\frac {dE}{dt} = -k Q(t) ## Then the rate at which the glucose in the body changes is: A) ## Q'(t) = \frac{dI}{dt} + \frac {dE}{dt} = r - k...
  17. Celso

    Differential equation for the simple pendulum

    How do I start this? I plugged the differential equation at wolfram alpha and it semmed too complicated for such an exercise. I've also looked at a sample of an answer on cheeg where the initial approach is to rewrite the equation as ##\frac{d}{dt} (\frac{\dot\theta^2}{2}-cos(\theta)) = 0## How...
  18. Phys pilot

    I How do I classify this partial differential equation? Inhomogeneous?

    Hello, I have to solve this second order differential equation. It's like a string vibrating equation but with a constant c: $$\frac{{\partial^2 u}}{{\partial t^2}}=k\frac{{\partial^2 u}}{{\partial x^2}}+c$$ B.C $$u(0,t)=0$$ $$u(1,t)=2c_0$$ c_0 is also a constant I.C $$u(x,0)=c_0(1-\cos\pi...
  19. cookiemnstr510510

    Differential Equations-Method of Undetermined Coefficients

    So to answer my question I need to reference another problem, I hope i won't get flagged for this... it is only to make a point about the way I am trying to approach this current problem. the previous problem stated: y''+2y'-y=10 so first I am finding Y_(homogeneous) and going straight to the...
  20. cookiemnstr510510

    Differential Equations-linear dependence of homogenous equations/genEq

    Here is my attempt at the solution: Y1(t)=kY2(t)→e^(3t)=ke^(-4t)→(e^(3t))/(e^(-4t))=k→e^(7t)=k So I have found a constant multiple of Y2(t), its the whole "interval" part that I don't get. The interval is (0,1), I guess I don't really know what they are trying to say...are they saying from 0...
  21. cdamberg21

    B Can an Impossible Differential Be Solved in Calculus?

    Hey, someone I know told me that the differential dy/dx= 24x/(2x+3) is not possible to solve... Is this true? If not what's the differential for it. This is my first year of calc in high school so my apologies if I butchered some of the terminology.
  22. AbusesDimensAnalysis

    A Differential equation involving a time series

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  23. Boltzman Oscillation

    Help derive this differential equation?

    Hello I need to derive this equation from Grittfith's quantum book $$ \frac{d^2y}{dr^2} = r^2y$$ I know I can use the characteristic equation: $$m^2 = r^2 \rightarrow y = e^{r^2}$$ but the answer should be: $$y=Ae^{\frac{-r^2}{2}} + Be^{\frac{r^2}{2}}$$ I know from Euler's formula that...
  24. D

    MATLAB Partial Differential Equation - solve with Matlab

    Help please, I need to solve this differential equation x\frac{\partial^2 U}{\partial x^2}+y\frac{\partial^2 U}{\partial y^2}=aU in Matlab (where "a" is a constant parameter, it can be taken by any), I wanted to use the Partial Differential Equation Toolbox, but I ran into a problem, the...
  25. F

    I Separation of Variables for Partial Differential Equations

    When using the separation of variable for partial differential equations, we assume the solution takes the form u(x,t) = v(x)*g(t). What is the justification for this?
  26. K

    Courses Taking Intro to Differential Equations after a break from school

    I'm currently an undergrad student who had to take a break from school for over a year and its been around 3 or 4 years since I took Calc I - Calc III and Linear Algebra. I'm debating on taking a introduction to differential equations course as an elective that starts in a couple weeks when I...
  27. echomochi

    Finding an implicit solution to this differential equation

    Homework Statement Find an equation that defines IMPLICITLY the parameterized family of solutions y(x) of the differential equation: 5xy dy/dx = x2 + y2 Homework Equations y=ux dy/dx = u+xdu/dx C as a constant of integration The Attempt at a Solution I saw a similar D.E. solved using the y=ux...
  28. S

    How to Integrate a Differential Equation to Find u?

    Homework Statement integrate: ##-v(du/dy) = κ(d^2(u)/dy^2) ## to obtain: ## (-v/κ)y = ln(du/dy) + c## and finally: ##u = d + w*e^(-vy/κ)## Homework Equations ##-v(du/dy) = κ(d^2(u)/dy^2) ## ## (-v/κ)y = ln(du/dy) + c## The Attempt at a Solution ## (-v/κ)dy = d(u) ## which gives: ##...
  29. R

    How do you solve a differential equation with complex numbers?

    Homework Statement Homework Equations euler ##e^{ix} = cos(x) + i*sin(x)## ##e^{-ix} = cos(x) - i*sin(x)## The Attempt at a Solution I'm starting with differential equations and I'm trying to understand this solution including complex numbers: First we determine the zeros. I understand that...
  30. Math Amateur

    I Differential Forms.... Another question.... Browder, Sec 13.1

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am reading Chapter 13: Differential Forms ... ... and am currently focused on Section 13.1 Tensor Fields ... I need some help in order to fully understand some statements by Browder in Section 13.1 ... ...
  31. Math Amateur

    I Differential Forms & Tensor Fiekds .... Browder, Section 13.1

    Andrew Browder in his book: "Mathematical Analysis: An Introduction" ... ... defines a differential form in Section 13.1 which reads as follows: In the above text from Browder we read the following: " ... ... A differential form of degree ##r## (or briefly an ##r##-form) in ##U## is a map...
  32. V

    Working with differential equations to obtain a function

    Homework Statement On a certain island, there is a population of snakes, foxes, hawks and mice. Their populations at time t are given by s(t),  f (t), h(t), and m(t) respectively. The populations grow at rates given by the differential equations s'=(8/3)s - f - (1/3)h - (1/6)m f'=(2/3)s + f -...
  33. A

    B Differential cross sections in Pythia

    Dear Users, I would like to ask you how can I plot d_sigma/d_omega and d_sigma/d_theta for any collision (for instance, proton and proton) using pythia event generator. I would be greatful if you could tell me how make it. Any ideas would be appreciated. Kind regards.
  34. BesselEquation

    Homogeneous Differential Equation

    Homework Statement Solve the following differential equation: y' = y / [ x + √(y^2 - xy)] 2. The attempt at a solution Using the standard method for solving homogeneous equations, setting u = y/x, I arrive at the following: ± dx/x = [1±√(u^2-u) ]/ [u√(u^2-u)] which in turn, I get the...
  35. Q

    I Deriving the spherical volume element

    I’m trying to derive the infinitesimal volume element in spherical coordinates. Obviously there are several ways to do this. The way I was attempting it was to start with the cartesian volume element, dxdydz, and transform it using $$dxdydz = \left (\frac{\partial x}{\partial r}dr +...
  36. confused student

    System of Differential Equations

    Homework Statement (It should be noted that the actual problem has specific values associated with a, b, and c. However, at this point I'm trying to find a method to solve the problem rather than a specific solution). Homework Equations The Attempt at a Solution When I was trying to solve...
  37. Z

    Differential Equation with an Initial condition

    Homework Statement x(dy/dx) = 3y +x4cos(x), y(2pi)=0 Homework Equations N/A The Attempt at a Solution I've tried a couple different ways to make this separable, but you always carry over a 1/dx or 1/dy term and I can never fully separate this. I've also tried to do a Bernoulli differential...
  38. M

    I Differential cross-sectional area - Rutherford's experiment

    I'm struggling to understand the importance of the differential cross-sectional area in Rutherford's scattering experiment, dσ/dθ. In one part of my course notes it is explained as 'the number of scatterings between θ and θ + dθ per unit flux, per unit range of angle'. However, dσ itself is...
  39. Math Amateur

    MHB The Dual Space and Differential Forms .... ....

    I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Chapter 8, Section 2, Differential Forms ... I need some help in order to fully understand some statements of Shifrin at the start of Chapter 8, Section 2 on the dual space ... The relevant text from...
  40. bob14

    Solving a Second-Order Nonlinear Differential Equation

    Homework Statement Hi, I'm trying to calculate the formula for the position vs. time of a rocket landing from an altitude of 100km. I'm neglecting a lot of forces for simplification but basically, I want to solve ##F_{net} = Drag - mg##. Homework Equations Drag Force: D = ## \frac {C_dAρv^2}...
  41. J

    B Why the Fourier series doesn't work to solve any differential equation?

    I know this may sound as a stupid question but I would like to clarify this. An arbitrary function f can be expressed in the Fourier base of sines and cosines. My question is, Can this method be used to solve any differential equation? You plug into the unkown function the infinite series and...
  42. Abhishek11235

    Is Every Differential 1-Form on a Line the Differential of Some Function?

    Homework Statement This problem is from V.I Arnold's book Mathematics of Classical Mechanics. Q) Show that every differential 1-form on line is differential of some function Homework Equations The differential of any function is $$df_{x}(\psi): TM_{x} \rightarrow R$$ The Attempt at a Solution...
  43. Peter Alexander

    Differential Nonlinearity: What happens when DNL = -1LSB

    Homework Statement Question is simple: what happens when ##DNL = -1 LSB## where DNL signifies differential nonlinearity and LSB stands for Least Significant Bit. It is also required to try and sketch such condition. Homework Equations Equation for differential nonlinearity: $$DNL(i) =...
  44. A

    Can You Solve These Differential Equations with Constants and Functions?

    Hi! I need your help for solving a couple of differential equation: dX/dt = a - b*X dY/dt = b*(c*exp(-E) - Y) - d*exp(-E)*Y X = X0 + f(Y, E) with X0, a, b, c and d are constants and f, a function of Z and E. Thank you in advance
  45. N

    Partial Differential Equation with variable coefficients

    Homework Statement This question relates to a very large project I have been assigned in a course on mathematical methods in structural engineering. I have to solve the following equation, in a specific way: (17) Now we have to assume the following solution: (18) It wants me insert...
  46. SpaceIsCool

    Can This Differential Equation Be Solved by Separation of Variables?

    Homework Statement We solved the differential equation (2.29), , for the velocity of an object falling through air, by inspection---a most respectable way of solving differential equations. Nevertheless, one would sometimes like a more systematic method, and here is one. Rewrite the equation...
  47. M

    A Laplace or Fourier Transform to solve a system of partial differential equations in thermoelasticity

    I've a system of partial diff. eqs. in thermo-elasticity, I can solve it using normal mode analysis method but I need to solve it using laplace or Fourier
  48. K

    Integral of a differential form

    Homework Statement Suppose that a smooth differential ##n-1##-form ##\omega## on ##\mathbb{R}^n## is ##0## outside of a ball of radius ##R##. Show that $$ \int_{\mathbb{R}^n} d\omega = 0. $$ Homework Equations [/B] $$\oint_{\partial K} \omega = \int_K d\omega$$ The Attempt at a Solution...
  49. A

    MHB Differential Equations - Bernoulli equation

    Afternoon, anyone that would like to take a look at this Differential Equation problem it would be very helpful. I have tried separating the problem, but I am only working with one known term. Consider the logistic equation $$\dot{y}=y(1-y). $$ (a) Find the solution satisfying $y_1(0)=6$ and...
  50. Baconslider

    Is This the Correct Approach to Solving the Exact Differential Equation?

    Homework Statement 2(y^2+1)dx+(4xy-3y^2)dy=0 Homework EquationsThe Attempt at a Solution here is my attempt to solve [/B]
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