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. Dusty912

    How Can Phasors Represent the Function g(t) in Complex Form?

    Homework Statement Given a function g(t)=acosωt + bsinωt, where a and b are constants, show that g(t) is the real part of the complex function: keiΦeiωt for some k and Φ Remark: the complex expression keiΦ is called a phasor. If we know that g(t) has the form kcos(ωt+Φ) then we need know only...
  2. Dusty912

    Match second-order differential equations with their y(t) graph

    Homework Statement The problem is in the picture. #17 I would have typed it but there are graphs that are needed for solving it. Basically trying to figure out which y(t) graph belongs to one of the sinusoidal forced equations with various parameters. Homework Equations euler's formula...
  3. D

    System of Differential Equations

    Homework Statement dx/dt + dy/dt = 2x + 2y + 1 dx/dt + 2(dy/dt) = y + 3 Homework Equations The Attempt at a Solution Dx + Dy = 2x + 2y + 1 Dx + 2Dy = y + 3 --Rearranging---------------------------- Dx - 2x = -Dy + 2y + 1 Dx = -2Dy + y + 3 ---Factoring----------------------------- (D...
  4. G

    Expressing phase space differential in terms of COM

    Homework Statement The Hamiltonian for a single diatomic molecule of identical atoms is given as $$H=\dfrac{\vec{p_1}\cdot\vec{p_1}}{2m}+\dfrac{\vec{p_2}\cdot\vec{p_2}}{2m}+\dfrac{K}{2}(\vec{r_1}-\vec{r_2})\cdot(\vec{r_1}-\vec{r_2})$$. Find the grand canonical partition function for a gas of...
  5. E

    A Nonhomogeneous second order nonlinear differential equations

    Hello every one, I have an equation related to my research. I wonder if anyone has any suggestion about solving it? y''+y' f(y)+g(y)=h(x) thanks
  6. R

    I Differential number of particles in Fermi gas model

    I'm practicing for the Physics GRE, and came across a question that has me stumped. "In elementary nuclear physics, we learn about the Fermi gas model of the nucleus. The Fermi energy for normal nuclear density (ρ0) is 38.4 MeV. Suppose that the nucleus is compressed, for example in a heavy ion...
  7. The Bill

    Analysis General texts on systems of partial differential equations?

    What are some good general textbooks on the properties and solution of systems of partial differential equations? I'm most interested in the general theory of vector and tensor valued PDEs like Maxwell's, Navier-Stokes, and the bulk equations governing elasticity and deformation of solids, etc...
  8. D

    I Interior product with differential forms

    Hi. I'm trying to self-study differential geometry and have come across interior products of vectors and differential forms. I will use brackets to show the interior product and I would just like to check I am understanding something correctly. Do I need to manipulate the differential form to...
  9. NihalRi

    Simple differential equation problem

    Homework Statement Find constants a and b such that y=ax + b is a solution to the differential equation dy/dx = 4x - 2y Homework EquationsThe Attempt at a Solution I already have the solution that is: a=4x-2 (ax+b) (I'm fine with this part) a = (4-2a) x - 2b (what happened here?) 4-2a=0...
  10. E

    I Question about using matrices for differential equations

    Let x(t)= [x1(t) x2(t)] be a solution to the system of differential equations: x′1(t)=−2x1(t)+2x2(t) x′2(t)==−6x1(t)+9x2(t) If x(0)= [4 -2] find x(t). I got the eigenvalues to be -6 and -5, but I don't know how to calculate the coefficients in front of the exponents. For lambda=-6 I get...
  11. Donghyun Kim

    A Is it possible to solve this differential equation?

    Hello. I'm solving the second order nonlinear ODE, and I'm not sure that it's possible or not. Please help. A_0, B_0, C_0, a, b, c, e, p is all known constant. dA/dt = -(a+b*C)*B dB/dt = -p*c*A dC/dt = -(1-p)*e*A then I want to get a solution like A(t) = function of (A_0, B_0, C_0, a, b, c...
  12. Cincin

    Difficult integral - Falling mass

    Homework Statement Solve the differential equation, dt/dv= 1/g (1/1-a^2*v^2) where a = (k/mg)^1/2 to yield v= 1/a(1-e^-2agt/1+e^-2agt). Homework Equations F=ma Newton's 2nd Law Integration Laws The Attempt at a Solution See image. I think I'm getting messed up on the integration laws.
  13. E

    Linear Differential equation problem

    Homework Statement Solution of the differential equation ##(\cos x )dy = y (\sin x - y) dx , 0 < x < \dfrac{\pi}{2} ## is Homework EquationsThe Attempt at a Solution Only separation of variables, homogenous and linear DEs are in the syllabus, therefore it must be one of those. It obviously...
  14. G

    I Symbol used in total differential and small errors

    Mod note: Thread title changed from "Sign used in total differential and small errors" to "Symbol used in total differential and small errors" My notes say that z=f(x,y) can be defined as the sum of the changes with respect to x,y respectively. The formula is: dz = ∂f/∂x * dz + ∂f/∂y * dy...
  15. R

    Designing a High-Gain Differential Amplifier: How to Meet Specific Requirements

    The figure shows a differential amplifier with a differential input vid = (vin_d_pos - vin_d_neg) between the bases of Q1 and Q2, and an output (out) at the emitter of Q4. I have to Determine R1, R2, R4, R5 and R6 so that the following requirements are met: R 1 = R 2 Av = vout / vid = 100 [V /...
  16. Geologist180

    What Is g'(2) for the Function G = (1/f^-1)?

    Homework Statement Suppose that f has an inverse and f(-4)=2, f '(-4)=2/5. If G= (1/f-1) what is g '(2) ? If it helps the answer is (-5/32) Homework Equations [/B] f-1'(b)=1/(f')(a) The Attempt at a Solution Im not really sure how to start this problem. I am familiar with how to use the...
  17. V

    I How do you integrate a term with a differential?

    In my Halliday Resnick Walker book, they derive the equations of kinematics using integrals in a small section after the algebraic derivation. I'm extremely confused by the calculus, however. Sorry if my questions are unclear.They say that \frac{dx}{dt}=v⇒dx=v\cdot dt. This first step itself...
  18. H

    B Recast a differential equation with a change of variable

    I obtained an extra factor of ##\frac{1}{l^2}## in the first term on the LHS of (3.34). From (3.33), LHS ##=u^2\frac{d}{d\theta}(\frac{u^2}{m}\frac{du}{d\theta}\frac{dr}{du})-\frac{l^2u^3}{m}## ##=u^2\frac{d}{d\theta}(-\frac{1}{m}\frac{du}{d\theta})-\frac{l^2u^3}{m}##...
  19. F

    Form differential equation by substituition

    Homework Statement dy/dx = (x-y+2) / (x+y-2) , by using x=X and y=Y+2 , form a differential equation . Homework EquationsThe Attempt at a Solution the author gave the answer as (Y^2) +(2XY) -(X^2)+A= 0 , while my answer is (Y^2) +(2XY) -(X^2)+A(X^4)= 0 is there anything wrong with my answer?
  20. Mrinmoy Naskar

    A What is the significance of the constant m in the Hermite differential equation?

    y" - 2xy' + my = 0 this is well known hermite diff eqn. now can anyone tell me what kind of conts is m?? what is the suitable value of m??
  21. Dusty912

    Differential Equations: linearity principle

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  22. A

    I Conceptual Question: Vector-Matrix Differential Equation

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  23. P

    B Are differential angles vectors?

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  24. S

    A Nonlinear first order Differential equation

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  25. G

    MHB System of differential equations involving matrices

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  26. chwala

    Differential equation tree height

    1. Homework Statement A tree is planted as a seedling of negligible height. The rate of increase of its height , in metres per year is given by ##0.2√(25-h)## a. explain why tree can't exceed 25 metres. answer⇒ ##dh/dt=0## when h=25 b. express t as a function of h answer⇒...
  27. A

    Solving Differential Equation numerically

    Homework Statement I am supposed to write a script that can solve the Schrödinger equation on a nonuniform grid. Homework Equations Finite element approximation to the second derivative as in: https://www.physicsforums.com/threads/nonuniform-finite-element-method.857334/#post-5382329 The...
  28. chwala

    A problem on differential equations

    Homework Statement A boy is eating a 250 gram burger. When he has eaten a mass m grams, his rate of consumption is given by ## 100-1/900m^2## grams per minute. How long does it take him to finish the meal?[/B]Homework EquationsThe Attempt at a Solution ##dm/dt=100-1/900m^2 ...
  29. G

    MHB Differential and integral equation.

    Hello! How do I solve the following integral and differential equations 1. $x^2f''(x)-2xf'(x) = x$ 2. $\int_0^{x} (1+f(x))\;{dx} = x$ I'm supposed to http://mathhelpboards.com/linear-abstract-algebra-14/linear-subspaces-17957.html all the polynomials that satisfy this, but I can't. (Shake)
  30. G

    General solution to differential equation

    Homework Statement Find the general solution to differential equation y''-6y'+9y=e^x((2x+1)\cos x+(x+3)\sin x) Homework Equations -Non homogeneous differential equation -Homogeneous differential equation with constant coefficients -Method of undetermined coefficients The Attempt at a Solution...
  31. G

    Dy/dx = (y^2) /x , form differential equation

    Homework Statement my ans is lnx = (-1/y) + c (-1/y) = lnx -c y = -1/ (lnx -c ) , but the answer given is (-1/ln x )+ C , how to get the answer given ? Homework EquationsThe Attempt at a Solution
  32. R

    Dirac Delta Function in Differential Equation

    Homework Statement Find the solution to: $$\frac{d^2}{dt^2} x + \omega^2 x = \delta (t)$$ Given the initial condition that ##x=0## for ##t<0##. First find the general solution to ##t>0## and ##t<0##. Homework Equations The Attempt at a Solution This looks like a non-homogeneous second...
  33. L

    (Tricky) Comet Picking up Mass - Differential Equations

    Homework Statement A comet in deep space picks up mass as it travels through a large stationary dust cloud. It is subject to a gravitational force of magnitude Mf acting in the direction of its motion. When it entered the cloud, the comet had mass M and speed V. After a time t, it has traveled...
  34. faradayscat

    Differential equations particular solution

    Homework Statement Particular solution of y" - y' - 2y = e^(2x) Homework Equations None The Attempt at a Solution This makes no sense to me, why do I have to use the solution of the form y(t) = cxe^(2x) For the problem above, but when I switch the signs and it becomes y" - y' + 2y =...
  35. B

    A Differential Equations not solvable

    In the first image it shows the ##\alpha^2-w_0^2<0## situation whereas in the second image the situation is when ##\alpha^2-w_0^2=0##.The problem is the book says to use ##T_0=2\pi/w_0## to determine diabetes but you can't do that when ##\alpha^2-w_0^2=0## because it can't be put into a cosine...
  36. T

    What Factors Determine the Performance of a BJT Differential Amplifier?

    Homework Statement Analyse the circuit to provide predictions for (a) the differential input resistance, (b) differential voltage gain, (c) single-ended voltage gain, (d) common-mode voltage gain, (e) common-mode rejection ratio (CMRR) in dB, (f) common-mode input resistance and (g) the...
  37. M

    Need help solving a differential equation

    Are there any known analytical method to solve the equation $$ A\frac{d^2f(x)}{dx^2}+B\frac{df(x)}{dx}+Ce^{igx}f(x) = 0\hspace{1cm}? $$ All quantities appearing in that equation are complex except for ##g## and ##x##.
  38. Z

    Is the differential in the momentum operator commutative?

    As it says; I was looking over some provided solutions to a problem set I was given and noticed that, in finding the expectation value for the momentum operator of a given wavefunction, the following (constants/irrelevant stuff taken out) happened in the integrand...
  39. Shackleford

    Derive differential equation that describes temperature

    I'm taking an engineering heat transfer course as an elective. 1. Homework Statement Copper tubing is joined to a solar collector plate of thickness t, and the working fluid maintains the temperature of the plate above the tubes at To. There is a uniform net radiation heat flux q”rad to the...
  40. H

    Differential equation of dy/dx = (2x-y+4)/ (4x-2y +1)

    Homework Statement I'm asked to use the transformation of v= 2x-y to solve dy/dx = (2x-y+4)/ (4x-2y +1) the answer given is (2/9)(6x-3y-2) +(7/9)ln(6x-3y-2) = x +c , i got (2/9)(6x-3y) +(7/9)ln(6x-3y-2) = x +c , what's wrong with my working ? Homework EquationsThe Attempt at a Solution
  41. H

    Automotive How do Center differentials work

    Greetings all, In my quest to imagine the ultimate sportscar, I began to wonder about the following: Certain cars are All Wheel Driven (AWD) and they do so via a Center differential delivering power to both axles. Amongst those cars, some have fixed Torque Bias/Split Ratio, which doesn't alter...
  42. F

    Solve the following differential equation

    Homework Statement Solve the differential equation. -x2(dy/dx) + xy = x2y2 * sin(x) Homework Equations None. The Attempt at a Solution I first figured out that this was a Bernoulli's equation. I distributed the x2 to make it simpler. From there I divided everything by y2 to get y-2(dy/dx) +...
  43. Y

    Impedance calculator for asymmetric differential stripline

    Hi Anyone know of an online asymmetric differential stripline impedance calculator? I already have the Saturn pcb design tool kit. I just want to find a second one to cross check the result. My experience is I have seen two calculators gave different results. Thanks
  44. H

    First order differential equation

    Homework Statement dy/dx = (x +y) / (x-y) , i am asked to find the first order differential equation , but the ans i gt is different from the ans given Homework EquationsThe Attempt at a Solution
  45. H

    How to Solve a Non-Separable Differential Equation?

    Homework Statement i was given dy/dx = -(4x+4y) / (4x+4y-2) , v= x +y . I have tried to do ( as in the photo) , but I didn't get the ans , in the last line of my working , the V and x are not separable , which part of my working is wrong ? Homework EquationsThe Attempt at a Solution
  46. T

    Differential equation help

    I understand what is in the picture http://postimg.org/image/u5ib33kzb/ but the book goes on to say that the solution is thus of the form ## X_n = a_n sin \frac{n \pi x}{l} ## How does putting ##β=σ^2=\frac{n^2π^2}{l^2}## into (6.37) result in that?
  47. C

    Why is differential equation equal to sum of partials?

    My current understanding of differential equations is extremely shaky, and my vocabulary is probably very incorrect, but I'm curious about something I've recently seen in some Khan Academy videos (specifically this one) and in other situations with differential equations. It seems that the...
  48. Math Amateur

    First approach to differential forms

    What do Physics Forums members regard as the best first introduction to differential forms ...?
  49. Math Amateur

    MHB First Approach to Differential Forms

    Can anyone suggest a good text or a good online set of notes from which to make a first approach to the topic of differential forms ... ? Similarly a first approach to to tensors ... ? The thought is to use these notions in order to gain an understanding of differential geometry and ... later...
  50. Dusty912

    Check My Work for solving this Differential Equation? :)

    Homework Statement Question: Solve this differential equation dy/dt=1+y2 and y(π/4)=-1 Homework Equations The Attempt at a Solution dy/dt=1+y2 and y(π/4)=-1 dy/dt=1+y2 dy=(1+y2)dt dy/(1+y2)=dt ∫dy(1+y2)=∫dt tan-1(y)=t+c y(t)=tan(t)+c y(π/4)=tan(π/4)+c -1=1+c -2=c answer: y(t)=tan(t)-2
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