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. F

    A Differential of Multiple Linear Regression

    Say you have a log-level regression as follows: $$\log Y = \beta_0 + \beta_1 X_1 + \beta_1 X_2 + \ldots + \beta_n X_n$$ We're trying come up with a meaningful interpretation for changes Y due to a change in some Xk. If we take the partial derivative with respect to Xk. we end up with...
  2. R

    MHB Bounded Solution For Differential Inequality

    Let x(t) a positive function satisfied the following differential inequality $\frac{x'(t)}{1+{x(t)}^{2}}+x(t)f(t)<2f(t)$ , (1) with $0\leq t\leq T$ , $\arctan(0)<\frac{\pi }{2}$ and $f(t)$ is a positive function. Is x(t) bounded for all $T\geq 0$?
  3. J

    Calculating pressures from flow rate & pressure differential

    It seems there must be a way, but I cannot seem to wrap my head around it. Here's the scenario... I have a 24" line flowing with a wet gas (combustion flue gas) into a dryer. It comes in at 90 degrees F, at a rate of 180 SCF/M. It leaves the dryer at 73 degrees F at a rate of 124...
  4. C

    Differential Equation for Projectile Motion with Air Drag

    Hi! I was wondering how I could come up with a differential equation for projectile motion on a 2D plane when air resistance is not negligible. I'm trying to guess the position of a projected ball at a certain time period by approximating the coordinates using the Euler's method. Here, I would...
  5. F

    Partial differential equation boundary

    Homework Statement I have to calculate the stationary field inside a room. Homework EquationsThe Attempt at a Solution I used the diffusion equation to calculate the temperature, which is T(x,y)=(Eeknx+Fe-knx)cos(kny), k=(n*pi/a), a is the length of the room. Now i have to satisfy boundary...
  6. ecoo

    Momentum Differential Equation

    Homework Statement A rocket sled moves along a horizontal plane, and is retarded by a friction force friction = μW, where μ is constant and W is the weight of the sled. The sled’s initial mass is M, and its rocket engine expels mass at constant rate dM/dt ≡ γ; the expelled mass has constant...
  7. F

    A Proving the Differential Map (Pushforward) is Well-Defined

    I am taking my first graduate math course and I am not really familiar with the thought process. My professor told us to think about how to prove that the differential map (pushforward) is well-defined. The map $$f:M\rightarrow N$$ is a smooth map, where ##M, N## are two smooth manifolds. If...
  8. G

    Help with this differential calculus

    <Moderator's note: Moved from a technical forum and therefore no template.> Hi everybody I've been trying to solve this problem all the afternoon but I haven't been able to do it, I've written what I think the answers are even though I don't know if they're correct, so I've come here in order...
  9. 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?
  10. 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 :=...
  11. 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...
  12. 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...
  13. 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?
  14. 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}...
  15. 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)...
  16. 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?
  17. 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...
  18. 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})...
  19. 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...
  20. 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.
  21. 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?
  22. W

    Is Substituting q(t) the Correct Method to Verify a Differential Equation?

    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...
  23. S

    I Problems in Differential geometry

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

    Partial differential wave (d'Alembert) solution check please

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  25. 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?
  26. 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...
  27. D

    Equilibrium volume of two differential van der Waal gases

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  28. 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...
  29. 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...
  30. 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.
  31. Saracen Rue

    Second Order Differential Equations - Beam Deflections

    Homework Statement A cantilever of length ##L## is rigidly fixed at one end and is horizontal in the unstrainted position. If a load is added at the free end of the beam, the downward deflection, ##y##, at a distance, ##x##, along the beam satisfies the differential equation...
  32. Saracen Rue

    B Second derivative differential equations in terms of y?

    Firstly I know how to do this with first derivatives in differential equations - for example say we had ##\frac{dy}{dx}=4y^2-y##, and we're also told that ##y=1## when ##x=0##. ##\frac{dy}{dx}=4y^2-y## ##\frac{dx}{dy}=\frac{1}{4y^2-y}=\frac{1}{y\left(4y-1\right)}=\frac{4}{4y-1}-\frac{1}{y}##...
  33. 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
  34. grquanti

    I Substitution in partial differential equation

    Hello everybody. Consider $$\frac{\partial}{\partial t}f(x) + ax\frac{\partial }{\partial x}f(x) = b x^2\frac{\partial^2}{\partial x^2}f(x)$$ This is the equation (19) of...
  35. S

    Differential Equation Resonance

    Homework Statement I was reading a PDE book with a problem of resonance $$ y_{tt} (x,t) = y_{xx} (x,t) + A \sin( \omega t) $$ After some work it arrived to a problem of variation of parameters for each odd eigenvalue. To solve it, it uses $$ y''(t)+a^{2} y(t) = b \sin ( \omega t) \qquad y(0)=0...
  36. K

    A A system of partial differential equations with complex vari

    Hi, I need to solve a system of first order partial differential equations with complex variables given by What software should I use for solving this problem..? The system includes 13 differential equations ...
  37. 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...
  38. 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...
  39. 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...
  40. 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...
  41. 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)...
  42. K

    How Accurate is Differential Approximation for Fourth Roots?

    Homework Statement Approximate ##~\sqrt[4]{17}~## by use of differential Homework Equations Differential: ##~dy=f(x)~dx## The Attempt at a Solution $$y=\sqrt[4]{x},~~dy=\frac{1}{4}x^{-3/4}=\frac{1}{4\sqrt[4]{x^3}}$$ $$\sqrt[4]{16}=2,~~dx=1,~~dy=\frac{1}{4\sqrt[4]{x^2}}\cdot 1=0.149$$...
  43. 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...
  44. 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...
  45. awholenumber

    I What are partial differential equations?

    If the slope of the curve (derivative) at a given point is a number .
  46. 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 $$2xy~dy+y^2~dx+x^2~dy+2xy~dx=0$$ $$x(2y+x)dy=-(y+2x)dx$$
  47. 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...
  48. 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...
  49. 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...
  50. 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...
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