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

    I Solving the differential equations involving SHM

    What is the most satisfactory explanation for guessing certain solutions to the differential equations encountered in damped & driven SHM?
  2. R

    Maple System of differential equations in Maple

    Hi everybody. I'm using the Maple 13 software (in linux mint) to solve system compounded by the four below differential equations: > ode1 := (diff(m1(t), t)) = - m1(t) + (1/2)*tanh( m2(t) + m4(t) + cos(t) ); > ode2 := (diff(m2(t), t)) = - m2(t) + (1/2)*tanh( m1(t) + cos(t) ); > ode3 :=...
  3. K

    Applied Differential geometry for Machine Learning

    My goal is to do research in Machine Learning (ML) and Reinforcement Learning (RL) in particular. The problem with my field is that it's hugely multidisciplinary and it's not entirely clear what one should study on the mathematical side apart from multivariable calculus, linear algebra...
  4. B

    Help with differential understanding

    Homework Statement A simple pendulum in quiet is released from the horizontal (##\theta=0##) (##\theta=90## for the vertical). Will the pendulum cover in the smaller time the arch from ##\theta=0## to ##\theta=30## or from ##\theta=30## to ##\theta=90##? The Attempt at a Solution I would like...
  5. S

    Wave properties from the differential equation of a wave

    How can we work out all the properties of wave from differential equation? And what really does differential equation of wave implies?
  6. H

    A Finite difference of fourth order partial differential

    What is a finite difference discretization for the fourth-order partial differential terms \frac{\partial u}{\partial x}k\frac{\partial u}{\partial x}\frac{\partial u}{\partial x}k(x,y)\frac{\partial u}{\partial x} and \frac{\partial u}{\partial x}k(x,y) \frac{\partial u}{\partial y}...
  7. A

    Help with mass-spring modeling problem

    Homework Statement Suppose at time zero, the bob was drawn upward four units from the equilibrium position, let C=2, K=2, m=1 lbm, initial speed=2 unit/sec find an expression for body's position. and in the solution it says: y''+6y'+5y=0 my question is: from where does the numbers (6) and (5)...
  8. S

    Engineering DC motor differential equation

    This question involves finding the transfer function for the system, but I first need to get the differential equations correct. Have I set up the gearbox correctly?
  9. A

    Solving partial differential equation with Laplace

    Homework Statement am trying to solve this PDE (as in the attached picture) https://i.imgur.com/JDSY4HA.jpg also my attempt is included, but i stopped in step, can you help me with it? appreciated, Homework EquationsThe Attempt at a Solution my attempt is the same as in the attached picture...
  10. Telemachus

    I Separation of variables for nonhomogeneous differential equation

    Hi. I was wondering if it is possible to apply separation of variables for a function of space and time obeying a non homogeneous differential equation. In particular, the heat equation: ##\displaystyle \frac{\partial \Phi(\mathbf{r},t)}{\partial t}-\nabla \cdot \left [ \kappa(\mathbf{r})...
  11. JTC

    A Split the differential and differential forms

    In undergraduate dynamics, they do things like this: -------------------- v = ds/dt a = dv/dt Then, from this, they construct: a ds = v dv And they use that to solve some problems. -------------------- Now I have read that it is NOT wise to treat the derivative like a fraction: it obliterates...
  12. F

    I Question about second order linear differential equations

    Hi everybody. I need to learn how to solve this kind of equation by decomposing y in a serie of functions. All the examples I have seen are of homogeneous functions. I would be extremely thankfull if someone pointed me to some text in which this is done-explained. Thanks for reading.
  13. M

    B Quantum vs. Classical Mechanics in Differential Element Analysis

    When we take a differential element for analysis why don't we consider quantum effects and only consider classical mechanics to solve the problem?
  14. W

    Differential Equations: LRC

    Homework Statement How does one show that q(t) is indeed a solution? Homework EquationsThe Attempt at a Solution My current idea is that i should come up with any form of solution, like q = Acos(ωt), and slot it in the RHS. Reason being that if q is indeed a solution, the result of the...
  15. S

    I Problems in Differential geometry

    Hello! Can someone point me toward some (introductory) problems in differential geometry with solutions (preferably free)? Thank you!
  16. C

    Partial differential wave (d'Alembert) solution check please

    Homework Statement Homework Equations General wave solution y=f(x+ct)+g(x-ct) [/B] The Attempt at a Solution [/B] Graphical sketch
  17. Gh. Soleimani

    A The differential equation of Damped Harmonic Oscillator

    If you consider b^2/m > 4*k, you can get the solution by using classic method (b = damping constant, m = mass and k = spring constant) otherwise you have to use complex numbers. How have the references books proved the solution for this differential equation?
  18. Marcus95

    Coupled differential equations using matrices

    Homework Statement We can treat the following coupled system of differential equations as an eigenvalue problem: ## 2 \frac{dy_1}{dt} = 2f_1 - 3y_1 + y_2 ## ## 2\frac{dy_2}{dt} = 2f_2 + y_1 -3y_2 ## ## \frac{dy_3}{dt} = f_3 - 4y_3 ## where f1, f2 and f3 is a set of time-dependent sources, and...
  19. D

    Equilibrium volume of two differential van der Waal gases

    Homework Statement Two ideal van der Waals fluids are contained in a cylinder, separated by an internal moveable piston. There is one mole of each fluid, and the two fluids have the same values of the van der Waals constants b and c; the respective values of the van der Waals constant ''a'' are...
  20. W

    Differential equations with eigenvalues.

    Homework Statement Find all solutions of the given differential equations: ## \frac{dx}{dt} = \begin{bmatrix} 6 & -3 \\ 2 & 1 \end{bmatrix} x ## Homework EquationsThe Attempt at a Solution So, we just take the determinate of A-I##\lambda## and set it equal to 0 to get the eigenvalues of 3...
  21. K

    B Derivation of exact differential

    Exact differential of a scalar function f takes the form of ∇f⋅dr=Σ∂ifdxi (where dr is a vector) f:R->Rnand I am not sure why this equation is valid in the sense that if we integrate the equation, ∫∇f⋅dr=∫{Σ∂ifdxi} ∫df=∫{Σ∂ifdxi} the above equation is true because integration is a linear...
  22. maistral

    A Integration by parts of a differential

    I'll cut the long story short. What on Earth happened here: I seem to be unable to do the integration by parts of the first term. I end up with a lot of dx's.
  23. Saracen Rue

    Second Order Differential Equations - Beam Deflections

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  24. Saracen Rue

    B Second derivative differential equations in terms of y?

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  25. K

    B Square root differential problem

    Hi, I working on their text this equation did not make sense to me. From equation 1 it differentiate second term , I wonder how he got second term of equation 2. What I think is, what I wrote at the bottom
  26. grquanti

    I Substitution in partial differential equation

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  27. S

    Differential Equation Resonance

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  28. K

    A A system of partial differential equations with complex vari

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  29. Salvador_

    Classical mechanics differential equation F(x) = -kx

    Homework Statement A particle of mass m is subject to a force F (x) = -kx. The initial position is zero, and the initial speed is v0. Find x(t). Homework Equations F = m*v*dv/dx = -kx v = dx/dt The Attempt at a Solution I'm new to differential equations, so please excuse me if I make any...
  30. Ron Burgundypants

    I Second order, non-linear, non-homogeneous differential eq.

    I have a physics project at university, we designed an experiment to measure the effectiveness of Poiseuilles law in a 'quasi non-steady state'. Poiseuilles law, simply being the measurement of the flow rate of a fluid in a pipe, holding only under steady state though. So by quasi steady state I...
  31. Drakkith

    I When is a Constant a Solution to a Differential Equation

    I've run across several instances while doing homework where a question will have two solutions. One will be an equation, and the 2nd will be a constant (usually zero). I can't figure out why this constant is a solution though. For example, take the following differential equation...
  32. 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...
  33. S

    Find a function that satisfies the following Differential Eq

    I'm helping some guys with Calculus I class and found this exercise in the practice about integrals. I think it's overkill but it may have some easy way to solve it. I'm very rusty solving differential equations. 1. Homework Statement Find f differentiable such that $$ (3+f'(x))e^{2-x} = (x-6)...
  34. K

    Approximation by differential

    Homework Statement Approximate ##~\sqrt[4]{17}~## by use of differential Homework Equations Differential: ##~dy=f(x)~dx## The Attempt at a Solution $$y=\sqrt[4]{x},~~dy=\frac{1}{4}x^{-3/4}=\frac{1}{4\sqrt[4]{x^3}}$$ $$\sqrt[4]{16}=2,~~dx=1,~~dy=\frac{1}{4\sqrt[4]{x^2}}\cdot 1=0.149$$...
  35. 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...
  36. J

    Geometry Differential Geometry book that emphasizes on visualization

    Hello! I would like to know if anybody here knows if there's any good book on academic-level dfferential geometry(of curves and surfaces preferably) that emphasizes on geometrical intuition(visualization)? For example, it would be great to have a technical textbook that explains the geometrical...
  37. awholenumber

    I What are partial differential equations?

    If the slope of the curve (derivative) at a given point is a number .
  38. K

    Differential of a y mixed with x

    Homework Statement Find dy of ##~xy^2+x^2y=4## Homework Equations Differential of a product: $$d(uv)=u\cdot dv+v\cdot du$$ The Attempt at a Solution $$2xy~dy+y^2~dx+x^2~dy+2xy~dx=0$$ $$x(2y+x)dy=-(y+2x)dx$$
  39. Drakkith

    I Quick Differential Form Question

    I've been going through my book learning about differential equations of multiple variables and I have a quick question about differential forms. If you are working a problem and get to the point where you're left with a differential form like ##(y)dx##, does that mean that the change in the...
  40. Drakkith

    I Question About Exact Differential Form

    My book is going through a proof on exact differential forms and the test to see if they're exact, and I'm lost on one part of it. It says: If $$M(x,y)dx + N(x,y)dy = \frac{\partial F}{\partial x}dx + \frac{\partial F}{\partial y}dy$$ then the calculus theorem concerning the equality of...
  41. Wrichik Basu

    Geometry Book Recommendations in Differential Geometry

    I wanted to study General Relativity, but when I started with it, I found that I must know tensor analysis and Differential geometry as prequisites, along with multivariable calculus. I already have books on tensors and multivariable calculus, but can anyone recommend me books on differential...
  42. 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...
  43. 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...
  44. S

    B Some help understanding integrals and calculus in general

    So in differential calculus we have the concept of the derivative and I can see why someone would want a derivative (to get rates of change). In integral calculus, there's the idea of a definite integral, which is defined as the area under the curve. Why would Newton or anyone be looking at the...
  45. Arman777

    Max values of current and charge using differential equations

    Homework Statement Homework Equations Circuit Equations. ##U_C=Q^2\2C## ##U_L=Li^2\2## The Attempt at a Solution For (a) I said ##100J## .But I think it might be ##200J## too.Here what I did; ##U_t=Q^2\2C## and I put ##Q=0.1C## and we know ##C##.Here I...
  46. A

    A Stochastic differential equations with time uncertainty....

    Hi all, I'm wondering if anyone is able to point me in a direction regarding an aspect of stochastic differential equations. I have a situation in which I need to propagate a stochastic DE through time using measurement updates - however, the exact time at which each measurement arrives is...
  47. 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...
  48. P

    I Differential forms as a basis for covariant antisym. tensors

    In a text I am reading (that I unfortunately can't find online) it says: "[...] differential forms should be thought of as the basis of the vector space of totally antisymmetric covariant tensors. Changing the usual basis dx^{\mu_1} \otimes ... \otimes dx^{\mu_n} with dx^{\mu_1} \wedge ...
  49. kupid

    MHB Few beginner doubts about differential equations ?

    I was trying to picture the third derivative of something Then i came across these ... What does displacement mean? The variable x is often used to represent the horizontal position. The variable y is often used to represent the vertical position Displacement=Delta x=xf-x0xf refers to the...
  50. 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...
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