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. cheapstrike

    Doubt related to formation of a differential equation

    Homework Statement Find the order of the differential equation of y=C1sin2x+C2cos2x+C3. Homework Equations - The Attempt at a Solution [/B] I read in my book that the order of the differential equation is equal to the number of arbitrary constants but the answer given is 2. Btw I have...
  2. R

    Differential Equation: x^2y''-xy'-3y=2x^-(3/2)

    Homework Statement Homework EquationsThe Attempt at a Solution I am not asking to find the answer, just wanted to know whether to use the variation of parameters or undetermined coefficients. Because this was on a test problem and I used variation of parameters instead. I know it is a...
  3. J

    I Surface parametrization and its differential

    I will use an example: -The surface is given by the intersection of the plane: y+z=2 -And the infinite cilinder: x2+y2<=1 We want to parametrize this surface, it could be done easily with: x=r cosθ y=r sin θ z=2 - r cos θ Then this surface could be written using vector notation: S= r...
  4. A

    Calculating the force for a differential pulley

    Homework Statement A differential pulley carries a weight W. The chain used has N links per foot. The bigger pulley has n notches, which can hold two chain links each. The smaller pulley contains n - 1 notches, which can also hold 2 chain links. The friction is such that the ratio of the force...
  5. C

    I Proofs of Stokes Theorem without Differential Forms

    Hello, does anyone have reference to(or care to write out) fully rigorous proof of Stokes theorem which does not reference Differential Forms? I'm reviewing some physics stuff and I want to relearn it. I honestly will never use the higher dimensional version but I still want to see a full proof...
  6. R

    Please help with this Differential equation problem

    Homework Statement y''-16y=2e^4x. Find general solution Homework EquationsThe Attempt at a Solution I have the homogenous equation which is c1e^-4x+c2e^4x, but I'm trying to find the particular solution for 2e^4x. I did yp=ae^4x, yp'=4ae^4x, yp''=16ae^4x, then plugged it into the equation...
  7. I

    System of differential equations in MATlab simulink

    Hello everyone, I would like if someone could help me with a little excersie here. 1. Homework Statement I am trying to simulate a mechanical system from a differential equations in simulink, but I don't know If I am doing it right. I've made the model as you can see in the 2nd picture...
  8. R

    Need help solving Differential Equation

    Sin(3x)dx+2ycos(3x)dy=0 So far, I have ∫2ydy=-∫sin(3x)/cos^2(3x)dx. Is that right? If so, how do I integrate sin(3x)/cos^2(3x)?
  9. T

    I About stochastic differential equation and probability density

    I have two questions about the use of stochastic differential equation and probability density function in physics, especially in statistical mechanics. a) I wonder if stochastic differential equation and PDF is an approximation to the actual random process or is it a law like Newton's second...
  10. J

    Partial differential coefficient

    Homework Statement The equation is z= e (x*y), the interesting thing is y is function of x too, y = ψ(x) Calculate the partial derivative respect to x, and the total derivative. Homework Equations Total differential: dz= ∂z/∂x dx + ∂z/∂y dy The Attempt at a Solution [/B] Well, according...
  11. H

    MHB Rigorous definition of "Differential"

    First of all I want to clarify that I posted this question on many forums and Q&A websites so the chances of getting an answer will be increased. So don't be surprised if you saw my post somewhere else. Now let's get started: When it comes to definitions, I will be very strict. Most textbooks...
  12. J

    Change of variables in a differential equation

    Homework Statement Transform the equation: x2 * d2y/dx2 + 2 * x * dy/dx + (a2/x2)*y = 0 Using: x=1/t Homework Equations The differential of a function of several variables, and the common rules of differentiation. https://en.wikipedia.org/wiki/Derivative The Attempt at a Solution As...
  13. Pouyan

    Fourier series and differential equations

    Homework Statement Find the values of the constant a for which the problem y''(t)+ay(t)=y(t+π), t∈ℝ, has a solution with period 2π which is not identically zero. Also determine all such solutions Homework Equations With help of Fourier series I know that : Cn(y''(t))= -n2*Cn(y(t)) Cn(y(t+π)) =...
  14. Telemachus

    Solving partial differential equation numerically

    Hi there. I am trying to self teach how to solve partial differential equations numerically using finite differences. I know this is a complex field, that requires much more knowledge of the theory than what I actually know, but anyway I wanted to try. Anyway, I've tried to build my own...
  15. P

    Matlab code problem with differential equations

    Homework Statement For a following differential equation d^2y/dx^2-4y=(e^x)/x  Find the solution using numerical methods Homework Equations d^2y/dx^2-4y=(e^x)/x The Attempt at a Solution %num dx=0.01; x=1:dx:3; l=zeros(1,length(x)); m=zeros(1,length(x)); l(1)=1; m(1)=0.25; for...
  16. Kernul

    Hooke's Law Second Order Differential Equation

    Homework Statement A mass ##m## on a frictionless table is connected to a spring with spring constant ##k## so that the force on it is ##F_x = -kx## where ##x## is the distance of the mass from its equilibrium position. It is then pulled so that the spring is stretched by a distance ##x## from...
  17. S

    Series solution for differential equation

    <OP warned about not using the homework template> Obtain a series solution of the differential equation x(x − 1)y" + [5x − 1]y' + 4y = 0Do I start by solving it normally then getting a series for the solution or assume y=power series differentiate then add up the series? I did the latter and...
  18. A

    MHB 2 Differential Equations by Substitution

    solve the following differential equation with the suggested change of variables.
  19. R

    I 2nd order differential equation problem with sin(theta)

    I have a differential equation that is essentially this: θ''(t)=c*sin[θ(t)] . I've been stymied trying to find a solution, and even when I tried using Maple, I got a nasty integral of a Jacobian amplitude. I'm tempted to use a small angle approximation, but the angle is 0≤θ≤π/2. I know this...
  20. S

    B How were differential equations for SIR Models calculated?

    Specifically:
  21. Mr Davis 97

    Construct a differential equation from the basis of solution

    Homework Statement Write down a 3x3 matrix A such that the equation ##\vec{y}'(t) = A \vec{y}(t)## has a basis of solutions ##y_1=(e^{-t},0,0),~~y_2 = (0,e^{2t},e^{2t}),~~y_3 = (0,1,-1)## Homework EquationsThe Attempt at a Solution I was thinking that, it looks like the matrix would have to...
  22. Cocoleia

    Non exact differential equation, initial value problem

    Homework Statement I am trying to solve the following: y'''-9y'=54x-9-20e^2x with y(0)=8, y'(0)=5, y''(0)=38 Homework EquationsThe Attempt at a Solution The right answer is: y= 2+2e^3x+2e^(-3x)-3x^2+x+2e^2x I am only wrong on the coefficients C2 and C3. Where did I mess up in my solution?
  23. S

    Unstable solution of differential equation

    Homework Statement Consider the following differential equation $$\frac{\partial^{2}\phi}{\partial t^{2}}-\nabla^{2}\phi = \phi(a-b\phi^{2}), \qquad a>0.$$ I would like to prove that $\phi=0$ is an unstable solution of this equation. Homework Equations The Attempt at a Solution Do you...
  24. Cocoleia

    Initial value problem - differential equations

    Homework Statement I am given (y^2 + y sin x cos y) dx + (xy + y cos x sin y) dy = 0, y(0) = π/2 . I need to solve this Homework EquationsThe Attempt at a Solution At this point they still aren't exact, so I gave up. I can't figure out what the problem is. Is it possible that I have to...
  25. Elvis 123456789

    Courses Partial Differential Equations vs Classical Mechanics 2?

    Hello everyone. So I wanted to get some opinions on what some of you thought was a better choice, as far taking PDE's or classical mechanics 2 goes. First let me start off by giving a little info; I've already taken calc 1-3 and ordinary differential equations, physics 1 & 2...
  26. G

    A Flatness Problem Differential Equation

    Hello to everybody, Since 3 days, I've been trying to obtain Eq. (173) from https://ned.ipac.caltech.edu/level5/Sept03/Trodden/Trodden5.html I know I need to derivate with respect to a, but it is being impossible to obtain the final answer. Can anybody help me with a detailed derivation...
  27. L

    B What is the Inverse of a Differential Operator?

    Hello everybody, If I define z_\mu = \frac{\partial{\phi}}{\partial{x^{\mu}}}, \, \mu = 0,1,...,n , (for some scalar function phi of x=(x_0,...,x_n)) how is then \frac{\partial{}}{\partial{z_{\mu}}} defined or rather what is it equal to? How would you call this expression? the inverse of a...
  28. J

    Studying Differential equations with complex functions?

    Hi folks, When you have a differential equation and the unknown function is complex, like in the Schrodinger equation, What methods should you use to solve it? I mean, there is a theory of complex functions, Laurent series, Cauchy integrals and so on, I guess if it would be possible to...
  29. W

    2nd Order Linear ODE-Derivation of system-issue

    Homework Statement How exactly they combined equation1 and equation2 and got that system? I don't get that part. Homework Equations A*(dy/dt)= -k*y eq1 A*(dz/dt)=ky-kz eq2 The Attempt at a Solution I tried substituting the 1st ky in the 2nd equation and then differentiating but I don't...
  30. resurgance2001

    I Euclidean differential number counts of supernovae

    Hi I am working on an assignment which is has asked us to derive an expression for a differential number count of supernovae in a euclidean flat non-expanding space. I am bit perplexed by this question and am wondering whether it is a trick question. We are allowed to do research to find an...
  31. W

    MHB Solve Diff. Eq. for g(x) w.r.t f(x) & c

    In the following equation: $$g'(x) = f'\left( x + \frac{c f'(x)}{\sqrt{ 1 + f'(x)^2 }} \right)$$ find $g(x)$ with respect to $f(x)$ where $c$ is any constant.
  32. T

    MHB So, the car takes approximately 8.43 seconds to travel 100 meters.

    Hi, another question I am having trouble with so my thought at the moment is to integrate a(t), which results in 1/2at^2 which is the kinematic equation for distance and then solve for the equation to equal 100. Just doesn't seem right to me and possibly too easy a solution.. think I am...
  33. T

    MHB Related rates and differential

    Hi, need some help on the following question. Just want to check on part a on the followingv=4/3\pi.r^3 dv = 4\pi.r^2 dr dv/dt = 4\pi.r^2 dr/dt dr/dt = (dv/dt)/ 4\pi.r^2 dr/dt = (-KA)/4\pi.r^2 dr/dt= -K part B need some help Thanks Tom
  34. T

    A How is this 'root stability' differential equation derived?

    I'm currently studying the sensitivity of polynomial roots as a function of coefficient errors. Essentially, small coefficient errors of high order polynomials can lead to dramatic errors in root locations. Referring to the Wilkinson polynomial wikipedia page right...
  35. Tspirit

    I How to solve the two following differential equations?

    (1) ##\frac{d^{2}y}{dx^{2}}=0## (2) ##\frac{d^{2}y}{dx^{2}}=k^{2}y##, where k is a real positive number.
  36. E

    Engineering Solving RLC circuit using differential equations

    Homework Statement Find the full response. Assume Vin is a squarewave with Vpp =10V and Vamp = +5V Homework Equations KCL The Attempt at a Solution My teacher gave this solution but I don't really understand some parts of it. Full response = Natural response + forced response Thevenin...
  37. G

    Modeling epidemics - solving differential equation

    I am given a modified SIR model in which the rate of decrease of susceptibles S is proportional to the number of susceptibles and the square-root of the number if infectives, I. If the number R of those who have been removed or recovered increases in proportion to the infectives, we have the...
  38. K

    Modeling differential equation

    Homework Statement Liquid is pouring into a container at a constant rate of 30cm^3s^-1 At time t seconds liquid is leaking from the container at a rate of 2/15 V cm^3s^-1, where V cm^3 is the volume of liquid in the container at that time. Show that -15 dV/dt = 2V - 450Homework Equations ...
  39. M

    A Nonlinear differential equation

    ρCp (∂T/∂t) + k (∂2T/∂x2) = exp(-σt2)exp(-λx2)φo i have this equation... i was thinking of taylor series expansion to solve it and make it easier... any ideas on how to solve?
  40. E

    Engineering Solve OpAmp Circuit Using Differential Equations

    Homework Statement Homework EquationsThe Attempt at a Solution Nodal Equations By property of OpAmp, V2=Vo eq1:\frac{V_{1}-V_{in}}{R_1}+\frac{V_{1}-V_{o}}{R_2}+C_2*(\dot{V_1}-\dot{Vo}) eq2: V_1=C_1R_2\dot{V_o}+V_o eq3: \dot{V_1}=C_1R_2\ddot{V_o}+\dot{V_o} Sub 2 & 3 into 1...
  41. dykuma

    Partial Differential equation, Temp in a Cylinder

    Homework Statement Homework Equations The Attempt at a Solution Because we are only looking at a cross section, I tried to reduce 5.3 down to just being a function of R and Theta. However I reasoned that there should be, based on this problem, no dependence on Theta either, so I figured I...
  42. Q

    Identifying Types of Singularity in Differential Equations

    Homework Statement Identify the type of singularity at x=0 for these differential equations x*Sin[1/x]*y''[x]+y[x]==0 x^2*y''[x]+Sin[1/x]*y[x]==0 Homework Equations A Singular point is regular if f(x)(x-x_0)^n is defined as x approaches x_0 and is analytic in a near a neighborhood of that...
  43. ShayanJ

    A Differential equations without Green functions

    Are there differential equations that, for some reason, don't have a Green function? Are there conditions for a DE to satisfy so that it can have a Green function? Thanks
  44. K

    I Differential Forms in General Relativity: Definition & Use

    Some time ago I was looking around the web for the use of differential equations in General Relativity. Then I found a definition (below) of differential forms, but I noted that the definition on my book is different from this one. Could someone tell me if it is right?
  45. H

    MHB Differential equations stability

    A one-dimensional dynamical system is given by $x′ = f(x), t \in [0,+\infty)$, where $f : \mathbb{R} \to \mathbb{R}$ is the smooth function defined as follows: $$f(x) = \begin{cases} x^4 \sin \left(\frac{1}{x}\right) & x \neq 0\\ 0 & x = 0. \end{cases}.$$ Find all the equilibrium points and...
  46. mr.tea

    I Constant solution and uniqueness of separable differential eq

    Hi, I am learning ODE and I have some problems that confuse me. In the textbook I am reading, it explains that if we have a separable ODE: ##x'=h(t)g(x(t))## then ##x=k## is the only constant solution iff ##x## is a root of ##g##. Moreover, it says "all other non-constant solutions are separated...
  47. F

    B Why differential of e^x is special?

    I learned that differential of e^x is same but what's so special about it? What makes is so special as it seems like a normal function to me other than the fact that e= sum of series of reciprocal of factorial numbers. What i want to ask is if e^x differential is e^x then do this rule apply to...
  48. L

    A Please help with 2-variable partial differential equation

    As a part of my research work, I need to find the number of charged particles at a given time 't', at a distance 'x' from anode. I derived a set of PDEs as per my requirement and assumptions which needs to be solved analytically. \begin{equation} \frac{\partial{N_e}}{\partial{t}} = \alpha N_e...
  49. M

    I How does the change in area compare to the differential area element?

    Hi PF! Suppose we have a differential area element ##dA##. This can be expressed as ##dx \, dy##. However, a change in area ##dA## seems different. Take positions ##x## and ##y## and displace them by ##dx## and ##dy## respectively. Then the change in area ##dA = (x+dx)(y+dy)-xy = xdy+ydx##...
  50. K

    I Differential Forms in GR: Higher Order Derivatives

    The differential form of a function is \partial{f(x^1,...,x^n)}=\frac{\partial{f(x^1,...,x^n)}}{\partial{x^1}}dx^1+...+\frac{\partial{f(x^1,...,x^n)}}{\partial{x^n}}dx^nIs there (especially in General Relativity) differential of higher orders, like \partial^2{f(x^1,...,x^n)}? If so, how is it...
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