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. K

    Basis of the solution space of a differential equation

    Homework Statement Verify that the functions y1(x) = x and y2(x) = 1/x are solutions of the differential equation y'' + (1/x)y' - (1/x2)y = 0 on I = (0,∞). Show that y1(x), y2(x) is a basis of the solution space of the differential equation. The Attempt at a Solution For the first part I'll...
  2. A

    Difficulty checking the solution of differential equation

    Homework Statement The differential equation is xy'-2y= x3ex Check the solution y=x2 ex Homework Equations Just plug in The Attempt at a Solution y' = x2ex + 2xex Having problem solving for x.
  3. X

    Research in Differential Geometry

    I am currently looking at grad schools, and I am wondering if anyone knew who are the leading researchers in differential geometry. I know that question is a little vague considering how diverse differential geometry is, but I was hoping that something could direct me in the right direction...
  4. ShayanJ

    Pair production differential cross section

    In one of my classes, I should give a talk about pair production cross section in front of the class and so I'm now searching for resources. But I can't find a place where the differential cross section for pair production process is given. Anyone knows somewhere I can find it and , preferably...
  5. Math Amateur

    MHB Differential Topology Notes - at undergraduate level

    I am trying to understand Differential Topology using several textbooks including Lee's book on Smooth Manifolds. I am looking for some good online lecture notes at undergraduate level (especially if they have good diagrams and examples) in order to supplement the texts ... Can anyone help in...
  6. C

    Small-signal transfer function from this differential input

    Homework Statement 2. Homework Equations [/B] Small Signal Equivalent Circuit, Kirchhoff Current Law, and BJT equations mentioned in the question. The Attempt at a Solution It looks like I did something wrong. Can someone help me with fixing this?
  7. E

    What is the advantage of using a differential probe for power calculation?

    Hi all, I found online that most professional websites suggest using differential probes to calculate power, along with a current probe, for a DUT. What advantage does the diff probe give us for power calculation? Thanks
  8. A

    Finding Solutions to Differential Equations with Constant Coefficients

    What is the general method for solving a differential equation of the form \begin{equation} \frac{\partial^{2}z}{\partial x^{2}}+\frac{\partial^{2}z}{\partial y\partial x}=C\end{equation} where C is a constant.
  9. N

    Differential equation (cannot separate)

    Homework Statement Solve for y using the substitution: z = 1/(y^5) dy/dx + y/x = (y^6)(x^3) Homework Equations (dz/dx) = (dz/dy) x (dy/dx) The Attempt at a Solution I formed an equation for dz/dx but cannot separate the variables in order to integrate. Can someone tell me where I've gone...
  10. P

    Hodge duality and differential forms

    If we have,$$A=d[(\bar{\alpha}-\alpha)(dt+\lambda)]$$ where $$\alpha$$ is a complex function and $$\lambda$$ is a 1-form. t here represents the time coordinate. If we want to calculate $$d\star A=0$$ where $$\star$$ is hodge star, we get if I did my calculations correctly...
  11. S

    Find the best T(t) from a differential eq. and conditions

    An electric vehicle of mass ##m## moves along the ##0-x## axis according to the law: m\frac{\mathrm{d} v(t)}{\mathrm{d} t} = T(t)-\mu mg-kv(t)^{2} (ma = Thrust - Friction - Drag) It is known that: P(t) = T(t)v(t) (Power = Thrust * Speed) Find the thrust ##T(t)## in such a way that the...
  12. E

    Vector differential identity proof (using triple product)

    Homework Statement Need to prove that: (v⋅∇)v=(1/2)∇(v⋅v)+(∇×v)×v Homework Equations Vector triple product (a×b)×c=-(c⋅b)a+(c⋅a)b The Attempt at a Solution I know I could prove that simply by applying definitions directly to both sides. I haven't done that because that is tedious, and I...
  13. N

    Help finishing a linear differential equation. Mechanics

    Homework Statement Find the distance which an object moves in time t if it starts from rest and has an acceleration d^2x/dt^2 = ge^-kt. Show that for small t the result is approx "x=(gt^2)/2" and show that for very large t, the speed is approximately constant. the constant is called the...
  14. M

    MHB Differential equation - Green's Theorem

    Hey! :o I want to find the solution of the following initial value problem: $$u_{tt}(x, t)-u_{xt}(x, t)=f(x, t), x \in \mathbb{R}, t>0 \\ u(x, 0)=0, x \in \mathbb{R} \\ u_t(x, 0)=0, x \in \mathbb{R}$$ using Green's theorem but I got stuck... I found the following example in my notes...
  15. evinda

    MHB Find solution of differential equation

    Hello! (Wave) The following differential equation is given: $$(1-x^2)y''-xy'+p^2y=0, p \in \mathbb{R}$$ Find the general solution of the differential equation at the interval $(-1,1)$ (with the method of power series). Are there solutions of the differential equation that are polynomials...
  16. B

    Wronskian to determine L.D

    Homework Statement Hello, I was just looking for a quick tip: If I have three distinct solutions to a second order linear homogeneous d.e, how would I show that the wronskian of (y1,y2,y3)(x)=0? I know how to show the wronskian is not zero for a linearly independent set, but I'm confused...
  17. D

    What is the interpretation of dx in calculus?

    Apologies if this isn't quite the right forum to post this in, but I was unsure between this and the calculus forum. Something that has always bothered me since first learning calculus is how to interpret dx, essentially, what does it "mean"? I understand that it doesn't make sense to consider...
  18. J

    Expressing the equation of hydrostatic equilibrium

    Homework Statement This problem has been stumping me for days now, I'm sure I'm missing something simple as it's only worth a small number of marks on the coursework. Any help would be appreciated. I've been asked to re-express the equation of hydrostatic equilibrium: dP/dr = - Gm/4πr4...
  19. B

    Is My Solution for a Non Homogeneous D.E. with Given Initial Conditions Valid?

    Homework Statement hello all, Suppose y is a solution of the d.e: y"+p(x)y'+q(x)y= q(x) on the interval (-1,1) with y(0)=1 and y'(0)= 1. What is y? Homework Equations I used the auxiliary equation: m^2+p(x)m+q(x)= q(x) The Attempt at a Solution My question is can I do this? I can cancel...
  20. D

    Calculate required torque for differential drive robot

    Hello everyone, So I am working on a differential drive vehicle... max weight 6kg and using two stepper motors centred on a circular shaped flat body. I want to try and work out if these stepper motors have enough torque to drive the thing. I am a second year engineering student, but I haven't...
  21. U

    Do i need differential equations?

    A friend recently gave me a book on quantum mechanics. It's called Introduction to quantum mechanics. It's by David j griffiths. I am currently taking multivariable calc.I am taking linear algebra next semester. I want study this book, but I am wondering what mathi I need. My friend told me I...
  22. Paradox101

    Radial Eigenfunction; Differential Equation

    Homework Statement Show that the radial eigenfunction unr,l is a solution of the differential equation: ħ2/2me×d2unr,l/dr2+[l(l+1)ħ2/2mer2 - e2/4πε0r]unr,l=Enr,lunr,lHomework Equations The radial function is R(r)=u(r)/r, so that the expression on the RHS is E×u. The Attempt at a Solution I know...
  23. T

    Help with First Order Differential

    Homework Statement The Attempt at a Solution The first part is fairly simple I think. It's just rate of accumulation = rate of generation - rate of output(losses) I'm not too sure how to solve this differential equation. I divide the whole equation through by mc and rearrange but I keep...
  24. patrickbotros

    Absolute Values in Separable Differential Equations

    When solving a separable differential equation, my textbook says this: ln|v-49|=-t/5+C→ |v-49|=e-t/5+C→ v=49+ce-t/5 What happened to the absolute values? I think it has something to do with the exponential always being positive.
  25. M

    How Can I Use the Slope Field Method to Solve a Differential Equation?

    Homework Statement dy/dx=2xcos(y)-xy^3[/B]Homework EquationsThe Attempt at a Solution dy/dx=2xcos(y)-xy^3=x(2cosy-y^3) dy/(2cosy-y^3)=xdx [/B] I can not integrate the left side of the equation. Can someone help me please?
  26. M

    Ordinary or partial differential equation

    Homework Statement x(d^2y/dx^2)+dx/dt+xy=0 Homework EquationsThe Attempt at a Solution At first I thought it was an ODE, but then I found out the derivative was respect to to variables x and t. I am not sure if it is an ODE or PDE. What are the dependent and independent variables in the...
  27. U

    2-form oriented triangle, Differential Forms

    Homework Statement Find the value of the 2-form dxdy+3dxdz on the oriented triangle with (0,0,0) (1,2,3) (1,4,0) in that order. Homework EquationsThe Attempt at a Solution I have tried various subtraction of these coordinates and applying them to the formula but the answer is in the back of...
  28. D

    Does the gear ratio in a differential affect pitching loads

    I'm designing a differential mount and am trying to wrap my head around the pitch loads the mount sees. My question is with a 1000 Ft/lb input torque will the ring and pinion ratio effect the torque that the mount must resolve? The differential is a sub-frame mounted IRS assembly, Nissan R230...
  29. T

    Differential Equations in simple mechanics?

    I came across a few problems in the Kleppner and Kolenkow book in which you must find the force of tension at specific lengths on a rope of mass m. They said you must use differential equations to solve these types of problems. How can you solve and use differential equations like this to get...
  30. W

    Differential cross-section for partonic collisions

    I am trying to calculate differential cross-section for partonic collisions (QCD) like q + q \rightarrow q +q q + \bar{q}\rightarrow q + q g + g \rightarrow g + g I can't find those calculations done anywhere, just the results and maybe some middle tips, that's all. As you may know those...
  31. B

    Differential equations and escape velocities

    Homework Statement Good day all! I'm stumped on a question: If I fire a bullet straight up what will be the initial velocity such that the bullet doesn't come back down? I need to model a differential equation (it will be first order) some how! Also, Gravity is not constant, but rather, the...
  32. evinda

    MHB Find solution that satifies the differential equations and has a specific form

    Hello! (Smile) I am looking at the following exercise: Let $I=(0,1)$. Find the solution $\phi$ that has a continuous derivative in $\mathbb{R}$ and satisfies : $$y''=0 \text{ in } I \\y''+k^2y=0 \text{ apart from } I, \text{ where } k>0$$ and furthermore $\phi$ has the form...
  33. A

    Applied Differential Equations in physics

    Hi everyone, First of all, this is an awesome place :) I'm looking for a differential equation book, with partial differential equation (and Fourier series solution) that really goes into physics. I'm a 3-year undergrad student in Physics so I already know a little about it. By the way, for now...
  34. M

    Unsteady state heat transfer differential equation

    I'm going through the solution to a problem that was assigned to my class and there's a step I don't really understand which I think is a concept I'm misunderstanding. 1. Homework Statement The curved surface of a cylinder of radius R and length L is insulated. The face at x = L is maintained...
  35. M

    Other Looking for a good differential equations text

    Good afternoon, I've been working my way through Serge Lang's series of textbooks, and I recently completed A First Course in Calculus. I'm currently working through the sequel to that book, Multivariable Calculus, and that should keep me tied up for at least two months. Looking ahead...
  36. C

    Linearity in differential equations

    Homework Statement Is the equation (x2sinx + 4y) dx + x dy=0 linear This problem also asks me to solve it, but I don't have a problem with that part. Homework Equations An equation is linear if the function or its derivative are only raised to the first power and not multiplied by each other...
  37. D

    Differential calculus, physics problem

    Homework Statement The motion of a spring that is subject to a frictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine function. Suppose the equation of motion of a point on such a spring is s(t) =...
  38. B

    Help Differential Equations types

    Homework Statement Good day all, My professor gave my class a packet of about 40 differential equations. I for the life of me cannot figure out how t solve these last 4! I also have an exam tomorrow morning, and would like finish these last few. I don't need them solved out, I would just...
  39. T

    Understanding Differential Equations: Exploring Relations between Functions

    Hello, I have a question that is relevant to differential equations. Say for example I have two functions that are related to one anothers derivatives. For example, the voltage acrossed an inductor is proportional to the rate of change of current through that inductor. My question for you is...
  40. O

    How to reduce higher order partial differential equations

    Hello guys, I have the system of PDE below and I want to solve it using finite difference method but I think I have to reduce it first to a system of first order PDE. The problem is that I don't know how to reduce this PDE to a first order system. I will appreciate any hints in this regard...
  41. L

    Nonlinear Ordinary Differential Equation Help

    Homework Statement y'=(x^2 +xy-y)/((x^2(y)) -2x^2)[/B]Homework EquationsThe Attempt at a Solution I know that really the only way to solve this one is to use an integrating factor, and make it into an exact equation. My DE teacher said that to make it into a exact equation you need to take...
  42. S

    MHB Second-order homogeneous linear differential equation

    Consider the second-order homogeneous linear differential equation $y'' + 4y' + Ky = 0$ Find the general solution if $K = 4$ So here is what I have: $r^2 + 4r + 4 = 0 $ =$(r + 2)(r+2)$ $r=-2$ ? But I thought that you can't do this because you won't be learning anything new if you have two of...
  43. stevendaryl

    How Do Spinors Fit in With Differential Geometry

    When I studied General Relativity using Misner, Thorne and Wheeler's "Gravitation", it was eye-opening to me to learn the geometric meanings of vectors, tensors, etc. The way such objects were taught in introductory physics classes were heavily dependent on coordinates: "A vector is a collection...
  44. PWiz

    Where do I start learning differential geometry?

    I've recently finished tackling differential equations. I want to start learning general relativity, but from what I've read, I need to have a firm footing in differential geometry first. So where do I start learning DG? I really don't want to do a half-hearted job in an attempt to quickly jump...
  45. evinda

    MHB Solution of differential equation

    Hello! (Wave) I am looking at the following exercise: We consider the differential equation $x^2y'+2xy+1=0$, where $0<x< +\infty$. Show that each solutions goes to $0$ while $x \to +\infty$. Find the solution $\phi$ of the above differential equation so that $\phi(2)=2 \phi(1)$. That's...
  46. resurgance2001

    Solving Simple Differential Equations: Particle Velocity and Position

    Homework Statement Given that a particle has an initial velocity v0 and then undergoes an acceleration a = - bv. , where b is a constant, obtain an expression for v = v(t) and x = x (t) [/B]Homework Equations Not sure The Attempt at a Solution If I integrate a = - b v I think I get v +...
  47. Dethrone

    MHB Differential Equation Challenge

    Find $y(x)$ to satisfy y(x)=y'(x)+\int e^{2x}y(x) \, dx+\lim_{{x}\to{-\infty}}y(x) given \lim_{{x}\to{0}}y(x)=0 and \lim_{{x}\to{\ln\left({\pi/2}\right)}}y(x)=1.
  48. L

    Second order differential equation

    Homework Statement So I'm in pchem right now and I haven't taken dif eq (it's not required, but I wish I had taken it now!) I am asked to solve this differential equation: y''+y'-2y=0 Homework Equations I know for a second order differential equation I can solve for the roots first. If...
  49. Logan Land

    MHB Need help setting up Differential Equation to solve

    dy/dx = (2y-x+5)/(2x-y-4) y(1)=1
  50. B

    Calculating Differential Mode Gain - CMRR, Acm, Adm

    Homework Statement http://i59.tinypic.com/1smqlu.png I'm not sure how to calculate differential mode gain Homework Equations I know formuals for CMRR and Acm but not Adm CMRR = | Adm/Acm | Rb/Ra = (1-ε)Rd/Rc
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