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. A

    Maple Is there a solution for these three differential equations?

    I have three differential equations with three unknowns ##p##, ##q## and ##r##: $$\displaystyle {\frac {\partial }{\partial p}}\sum _{k=1}^{5}f_{{k}}\ln \left( P \left( X=k \right) \right) =0$$, $$\displaystyle {\frac {\partial }{\partial q}}\sum _{k=1}^{5}f_{{k}}\ln \left( P \left( X=k...
  2. docnet

    System of two differential equations

    The first equation leads to x = ae^2t + be^t and the second equation leads to y=[1/(ln(sint+pi/2)+c)] this corresponds to the system a+b=1/c 2a+b=1 which has infinitely many solutions. what am I missing here?
  3. Vossi

    Electric Circuit Differential equation help

    I'm confused at the part how 4Vc and 48 cos(60t) are deduced, that's all.
  4. karush

    MHB General Solution of Differential Equation System

    $\tiny{27.1}$ 623 Find a general solution to the system of differential equations $\begin{array}{llrr}\displaystyle \textit{given} &y'_1=\ \ y_1+2y_2\\ &y'_2=3y_1+2y_2\\ \textit{solving } &A=\begin{pmatrix}1 &2\\3 &2\end{pmatrix}\\...
  5. chwala

    Solving this differential equation

    ##-\frac {dy}{dx}=\frac {3+4v}{4-3v}## ##\frac {3+4v}{4-3v}=-v-x\frac {dv}{dx}## ##-\frac {dx}{x}=\frac {4-3v}{8v-3v^2+3}## ##\frac {dx}{x}=\frac {4-3v}{3v^2-8v-3}##=[A/3v+1]+[B/v-3]##
  6. B

    Determine the new differential reading in a manometer

    Not sure how to start off this question I'm confused how to begin if I do not the exact pressure on either pipe A or pipe B Only thing that I can deduce from this is that if pipe A exerts a smaller pressure than before then the mercury column on the left side would rise i.e. the new...
  7. M

    MHB Natural and forced response of a differential equation

    Greetings everyone, I am a bit new to differential equations and I am trying to solve for the natural and forced response of this equation: dx/dt+4x=2sin(3t) ; x(0)=0 Now I know that for the natural response I set the right side of the equation equal to 0, so I get dx/dt+4x=0, thus the...
  8. nomadreid

    I Did von Neumann coin the eth or dyet for the inexact differential?

    Today the inexact differential is usually denoted with δ, but in a text by a Russian author I found a dyet (D-with stroke, crossed-D) instead: In response to my question to the author about this deviation from normal usage, he stated that this was a suggestion from von Neumann. (Which of course...
  9. O

    Problem in solving differential equation

    Hello everyone! I was studying chaotic systems and therefore made some computer simulations in python. I simulated the driven damped anhatmonic oscillator. The problem I am facing is with solving the differential equation for t=0s-200s. I used numpy.linspace(0,200,timesteps) for generate a time...
  10. PainterGuy

    Finding the position of a differential drive robot

    Data: The speed of right wheel is considered to be 25 RPM. The speed of left wheel is considered to be 20 RPM. The distance, L, between wheels is 30 cm. Also the radius, r, of each wheel is 6 cm. Question: Using the data above for a differential robot, find the following: i: angular...
  11. Amit1011

    What substitution/other method to use to solve this differential equation?

    Differentiating eq1 mentioned above, and using eq 2, i got : $$v\frac{dv}{d\theta}=R\frac{dv}{dt}$$ From this, i got:$$ \frac{d\theta}{dt}=\sqrt{(2/R)(g(1-cos\theta )+asin\theta)}$$ After this point, I am not able to understand what substitution or may be other method could be used to solve...
  12. S

    B Exploring Differential of Vector Component vs Change During Parallel Transport

    I'm reading 'Core Principles of Special and General Relativity' by Luscombe - the part on parallel transport. I guess ##U^{\beta}## and ##v## are vector fields instead of vectors as claimed in the quote. Till here I can understand, but then it's written: I want to clarify my understanding of...
  13. Ron Burgundypants

    Second order differential equation solution

    I know the solution to the equation (1) below can be written in terms of exponential functions or sin and cos as in (2). But I can't remember exactly how to get there using separation of variables. If I separate the quotient on the left and bring a Psi across, aka separation of variables (as I...
  14. Linder88

    Second order differential equation

    We choose an approximative solution given by $$ u_N(x) = \frac{a_0}{2} + \sum_{n=1}^N a_n \cos nx + b_n \sin nx $$ Comparing this approximative solution with the differential equation yields that $$ \frac{a_0}{2} = a $$ and the boundary conditions yields the equation system $$ a + \sum_{n=1}^N...
  15. M

    A Solving a differential equation with two unknowns

    One thing that is given in paper (attached) is a operating set point for temperature which is given as 20 for day and 16 for night but I do not know whether its initial condition for temperature or not. Can anyone please guide me that what kind of equation is it and how can I solve it with these...
  16. M

    MHB Null Cline Solutions for Differential Equations: Help Needed!

    I was wondering if anyone could help me clarify which null cline solutions are correct for this question I've got: I've got two differential equations: \[ du/dt =u(1-u)(a+u)-uv \] \[ dv/dt = buv-cv \] where a, b and c are constants. I know to find the u null clines you set du/dt to 0. \[...
  17. R

    A Compton Recoil and the Electron Differential Cross Section

    Hi folks, My understanding of the Compton Effect is that maximum energy transfer to the electron takes place when the photon scattering angle is 180 degrees. For the following please reference Evans "The Atomic Nucleus" ...
  18. madafo3435

    I Differential analysis: convex functions

    I am reading A Course in Mathematical Analysis Volume 1 by D. J. H. Garling, and I am having trouble in the following demonstration of Section 2 Differentiation. part 4 of the test, the first part of the second inequality does not make sense, I do not understand its justification. I hoped they...
  19. schofieldius

    Help with the equations for this "sideways differential" please

    looking for some assistance working out torque equations for this set-up please.
  20. agnimusayoti

    Differential equation on relativistic momentum (ML Boas)

    $$p=\gamma m v$$ $$F = \frac {md (\gamma v}{dt}$$ $$\int{F dt} = \int{md (\gamma v}$$ $$F t= \gamma mv$$ At this step, I don't know how to make v as explicit function of t, since gamma is a function of v too. Thankss
  21. K

    A Can we always rewrite a Tensor as a differential form?

    I read in the book Gravitation by Wheeler that "Any tensor can be completely symmetrized or antisymmetrized with an appropriate linear combination of itself and it's transpose (see page 83; also this is an exercise on page 86 Exercise 3.12). And in Topology, Geometry and Physics by Michio...
  22. K

    Geometry Modern Differential Geometry Textbook Recommendation

    Could you provide recommendations for a good modern introductory textbook on differential geometry, geared towards physicists. I know physicists and mathematicians do mathematics differently and I would like to see how it is done by a physicists standard. I have heard Chris Ishams “Modern Diff...
  23. A

    Rocket propulsion explained only through pressure differential

    Fluid can exert force to object(move object) only through pressure and tangential stress caused by viscosity. if we look at balloon rocket ,here is Newton 3 law action-reaction,but this 3 law as usual don't tell nothing how fluid really exert force to the ballon.. it exert through pressure...
  24. Kaguro

    Mixed method of solving differential equations

    I use the operator method here: (D^2 + D+3)y = 5cos(2x+3) ## y = \frac{1}{D^2+D+3} 5cos(2x+3) ## ## \Rightarrow y= \frac{5}{-(2)^2+D+3}cos(2x+3) ## ## \Rightarrow y= \frac{5}{-4+D+3}cos(2x+3) ## ## \Rightarrow y= \frac{5}{D-1}cos(2x+3) ## At this, if I revert back to write: (D-1)y = 5cos(2x+3)...
  25. S

    Solve this differential equation for the curve & tangent diagram

    Here is my attempt at a solution: y = f(x) yp - ym = dy/dx(xp-xm) ym = 0 yp = dy/dx(xp-xm) xm=ypdy/dx + xm xm is midpoint of OT xm = (ypdy/dx + xm) /2 Not sure where to go from there because the solution from the link uses with the midpoint of the points A and B intersecting the x-axis...
  26. Yellotherephysics

    A Functional Determinant of a system of differential operators?

    So in particular, how could the determinant of some general "operator" like $$ \begin{pmatrix} f(x) & \frac{d}{dx} \\ \frac{d}{dx} & g(x) \end{pmatrix} $$ with appropriate boundary conditions (especially fixed BC), be computed? And assuming that it diverges, would it be valid in a stationary...
  27. paulmdrdo

    Differential amplifier common mode gain

    Summary:: Differential amplifier common mode gain derivation of forumlas I'm having a hard time deriving for equations 10-8 -10-9. I tried adding equation's 18-6 and 18-7 but cannot proceed with the derivation. I need help on this. Thank you!
  28. T

    I Vector space for solutions of differential equations

    Good Morning Recently, I asked why there must be two possible solutions to a second order differential equation. I was very happy with the discussion and learned a lot -- thank you. In it, someone wrote: " It is a theorem in mathematics that the set of all functions that are solutions of a...
  29. C

    Engineering Calculate the max common mode voltage at the differential amplifier inputs

    Im unsure if I am on the correct track or have gone off on a tangent. Any help or guidance would be appreciated. CMRR=20log10(Adiff/Acm) 120=20log10(10^5/Acm) 120/20=log10(100,000/Acm) 6=log10(100,000/Acm) taking antilogs 1,000,000=100,000/Acm Acm=100,000/1,000,000 Acm=0.1Max amplified...
  30. S

    I Differential form of the velocity equation in a non-standard configuration

    I'm reading a text on special relativity (Core Principles of Special and General Relativity), in which we start with the equation for composition of velocities in non-standard configuration. Frame ##S'## velocity w.r.t. ##S## is ##\vec v##, and the velocity of some particle in ##S'## is ##\vec...
  31. K

    A Differential Forms or Tensors for Theoretical Physics Today

    There are a few different textbooks out there on differential geometry geared towards physics applications and also theoretical physics books which use a geometric approach. Yet they use different approaches sometimes. For example kip thrones book “modern classical physics” uses a tensor...
  32. mertcan

    A Solution to a differential equation with variable coefficients

    Hi, I really struggled to dig valuable things out of internet and books related to high order homogeneous differential equation with variable coefficients but I have nothing. All methods I see involves given solution and try to find others(like reduction of order method), even for second order...
  33. A

    A Differential Equations (Control Optimization Problem)

    \begin{equation} y_{1}{}'=y_1{}+y_{2} \end{equation} \begin{equation} y_{2}{}'=y_2{}+u \end{equation} build a control \begin{equation} u \epsilon L^{2} (0,1) \end{equation} for the care of the appropriate system solution \begin{equation} y_{1}(0)=y_{2}(0)=0 \end{equation} satisfy...
  34. A

    MHB Optimal Control for Differential Equations with L2 Control Constraint

    To be able to build a control y_{1}{}'=y_1{}+y_{2} y_{2}{}'=y_2{}+u u \epsilon L^{2} (0,1) for the care of the appropriate system solution y_{1}(0)=y_{2}(0)=0 satisfy y_{1}(1)=1 ,y_{2}(1)=0 Please kindly if you can help me Discipline is Optimal ControlHELP! i need to find...
  35. E

    B A differential equation, or an identity?

    This is quite literally a showerthought; a differential equation is a statement that holds for all ##x## within a specified domain, e.g. ##f''(x) + 5f'(x) + 6f(x) = 0##. So why is it called a differential equation, and not a differential identity? Perhaps because it only holds for a specific set...
  36. Tymofei

    Pressure Differential in Fluid Dynamics: Why More on the '+s' Side?

    Here i added a page from my fluid dynamics book where it shows particle model for deriving the equation. My question is why pressure is more at stream side aka 'positive "s" direction'.I would expected more pressure on the other side because for example when you trying to push a rigid object or...
  37. K

    I Differential equation with two terms

    I'm trying to solve a differential equation of the form $$\frac{A'(x)}{A(x)}f(x,y) = \frac{B'(y)}{B(y)}$$ where prime denotes differentiation. I know that for the case ##f(x,y) = \text{constant}## we just equal each side to a same constant. Can I do that also for the case where ##f(x,y)## is not...
  38. LCSphysicist

    Differential Pulley: Force to Balance Weight W

    I am trying to deal with this problem, the question is what is the force to balance the weight W, where the rope don't have weight. The bigger pulley at the top has radius a, and the other, attached to the same axis, has radius 0.9a. The force is applied in one side of the freeling rope. I...
  39. JD_PM

    A Understanding how to derive the differential cross-section formula

    Please let me make questions after showing what I am studying. We first consider two particles (they may be either leptons or photons) with initial (i.e. before collision) four momentum ##p_i = (E_i, \mathbf p_i)##, ##i=1,2##. These two collide and produce ##N## final particles with momentum...
  40. M

    A Partial Differential equation (Heat eqn)

  41. J

    Studying Ordinary Differential Equations and Calc III

    Hello, I need help deciding on whether to take ODE (MAP2302) and Calc III during the summer. Would it be wise to take ODE along with Calc III in the same semester? Some people have told me to take Calc III first because there are a few things in ODE that are taught in Calc III, but others have...
  42. SilverSoldier

    Differential Equation for a Pendulum

    Suppose we displace the pendulum bob ##A## an angle ##\theta_0## initially, and let go. This is equivalent to giving it an initial horizontal displacement of ##X## and an initial vertical displacement of ##Y##. Let ##Y## initially be a negative number, and ##X## initially be positive. I observe...
  43. E

    Solving an exact differential equation

    I let ##M = 4xy + 1## and ##N = 2x^2 + \cos{(y)}##. Since ##\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}##, the equation is exact and we have $$\frac{\partial f(x,y)}{\partial x} = 4xy + 1$$ From inspection, you can tell this has to lead to $$f(x,y) = 2x^2 y + x + h(y)$$ and we...
  44. Tony Hau

    I Deriving the differential equation for the underdamped case

    The formula for general oscillation is: The formula for underdamping oscillation is: where λ = -γ +- sqart(γ^2 - ω^2), whereas A+ and A- , as well as λ+ and λ-, are complex conjugates of each other. After some operations, we get: x(t) = Ae^(-γx)[e^i(θ+ωx) +e^-i(θ+ωx)], where A is the modulus...
  45. caffeinemachine

    MHB Maximum value a function satisfying a differential equation can achieve.

    Let $f:\mathbb R\to \mathbb R$ be a twice-differentiable function such that $f(x)+f^{\prime\prime}(x)=-x|\sin(x)|f'(x)$ for $x\geq 0$. Assume that $f(0)=-3$ and $f'(0)=4$. Then what is the maximum value that $f$ achieves on the positive real line? a) 4 b) 3 c) 5 d) Maximum value does not exist...
  46. C

    A Partial differential equation containing the Inverse Laplacian Operator

    I am trying to reproduce the results of a thesis that is 22 years old and I'm a bit stuck at solving the differential equations. Let's say you have the following equation $$\frac{\partial{\phi}}{\partial{t}}=f(\phi(r))\frac{{\nabla_x}^2{\nabla_y}^2}{{\nabla}^2}g(\phi(r))$$ where ##\phi,g,f## are...
  47. sophiatev

    I Invariance of Diff. Line Elems. in Hartle's Gravity: Intro to GR

    In Hartle's book Gravity: An Introduction to Einstein's General Relativity he spends chapter 2 discussing some basic aspects of differential geometry. For example, he derives the expression for a differential line element in 2D Euclidean space: dS^2 = (dx)^2 + (dy)^2 in Cartesian coordinates...
  48. snoopies622

    Arriving at the differential forms of Maxwell's equations

    In college I learned Maxwell's equations in the integral form, and I've never been perfectly clear on where the differential forms came from. For example, using \int _{S} and \int _{V} as surface and volume integrals respectively and \Sigma q as the total charge enclosed in the given...
  49. Ayoub Tamin

    Mathematica How to solve this differential equation using Mathematica's Dsolve?

  50. K

    B Differential equations in physics

    Can someone list to me (and whoever is going to view this thread) what topics in differential equations should be studied so that we can have a decent knowledge of the general physical theories in which they occur? (And I believe, they appear in all theories.) So far, I believe the two most...
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