Differential equations Definition and 999 Threads

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.

View More On Wikipedia.org
Recent contents Top users
  1. Linder88

    State-Space (SS) Formulation for Equations of Motion

    Homework Statement The task is to write the following equations of motion as in equation (2) considering the inputs and outputs as in equation (3) \begin{equation} \begin{cases}...
  2. E

    Solve coupled nonlinear differential equations

    Good evening I have these coupled equations and was wondering if there is any chance solving them analytically. If not, how would you approach it numerically? (shown in attachment) Thank you very much
  3. kostoglotov

    Diff eqs with eigenvectors: double roots, but 2nd eigenvector?

    The problem is here, I'm trying to solve (b): imgur link: http://i.imgur.com/ifVm57o.jpg and the text solution is here: imgur link: http://i.imgur.com/qxPuMpu.pngI understand why there is a term in there with cte^t, it's because the A matrix has double roots for the eigenvalues. What I...
  4. kostoglotov

    What is the solution for a system of ODEs with a matrix coefficient?

    I know how to solve \frac{d\vec{u}}{dt} = A\vec{u}, I was just watching a lecture, and the lecturer related that solving that equation is pretty much a direct analogy to \vec{u} = e^{At}\vec{u}(0), in so far as all we need to do after that is understand exactly what it means to take the...
  5. kostoglotov

    How can e^{Diag Matrix} not be an infinite series?

    So, in a section on applying Eigenvectors to Differential Equations (what a jump in the learning curve), I've encountered e^{At} \vec{u}(0) = \vec{u}(t) as a solution to certain differential equations, if we are considering the trial substitution y = e^{\lambda t} and solving for constant...
  6. N

    Find the approximate linear ODE system

    dx/dt = x-y^2 dy/dt= x^2 -xy -2x For each critical point, find the approximate linear OD system that is valid in a small neighborhood of it. I found the critical points which are (0,0),(4,2),(4,-2) but have no idea how to do the above question! please help!
  7. D

    Motion described by differential equations

    Homework Statement Text from a classical mechanics textbook ( uploaded picture ) shows 2 diff equation describing the motion graphically presented in the uploaded picture. How were these set up? Homework EquationsThe Attempt at a Solution I don't have a slightest clue as how are these...
  8. Aristotle

    Laplace Transform Method for Solving Initial Value Problems

    Homework Statement Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teacher is against using it..) y'' + 2y' + 2y = 2 ; y(0)= y'(0) = 0 Homework Equations Lf'' = ((s^2)*F) - s*f(0) - f'(0) Lf' = sF - f(0) Lf = F(s) The...
  9. M

    MHB Linear differential equations with constants coefficients

    Hey! :o Each element of the ring $\mathbb{C}[z, e^{\lambda z} \mid \lambda \in \mathbb{C}]$ is of the form $\displaystyle{\sum_{k=1}^n α_kz^{d_k}e^{β_kz}}$. A differential equation in this ring is of the form $$Ly = \sum_{k=0}^m \alpha_k y^{(k)}(z)=\sum_{l=1}^n C_lz^{d_l} e^{\beta_l z} , \ \...
  10. D

    Green's functions for translationally invariant systems

    As I understand it a Green's function ##G(x,y)## for a translationally invariant differential equation satisfies $$G(x+a,y+a)=G(x,y)\qquad\Rightarrow\qquad G(x,y)=G(x-y)$$ (where ##a## is an arbitrary constant shift.) My question is, given such a translationally invariant system, how does one...
  11. L

    SHM - as two ordinary linear differential equations

    Homework Statement I've attached an image of the problem question, it's Q1 I'm working on This is what I have so far: we have two components of SHM, position x and velocity v. when x = 0, v = a maximum, when v = 0, x = a maximum this is represented by sin & cos functions. where x =...
  12. Hutchyy

    Ordinary Differential equations question

    #17 If you can't see the picture: Suppose that y1, y2, and y3 are solutions to a third order constant coefficient homogeneous differential equation. Suppose further that for all real t, W(y1,y2)(t)>0, but also W(y1,y2,y3)(0)=0. Then there exists c1 and c2 such that c1y1(t) + c2y2(t) =y3(t) for...
  13. PhotonSSBM

    Application or Theoretical Differential Equations?

    There are two courses I can take for a Differential Equations class at my school. One is for Engineering students and is described this way (I'm a physics major fyi): This course presents an introduction to the theory of differential equations from an applied perspective. Topics include linear...
  14. ecoo

    Integrating differential equations that have ln

    Hey guys, I have a question concerning the rewriting of a differential equation solution. In the example above, they rewrite [y=(plus/minus)e^c*sqrt(x^2+4)] as [y=C*sqrt(x^2+4)]. I understand that the general solution we get as a result represents all the possible functions, but if we were to...
  15. Julio1

    MHB How Can You Reduce This PDE to Its Canonical Form?

    Let the PDE $u_{xx}-4u_{xy}+4u_{yy}=0.$ Reduce to the canonical form.Good Morning MHB :). My problem is find the canonical form of the PDE know an variable change. But how I can transform the equation? Thanks.
  16. S

    Solve Initial Value Problem and Determining Interval

    Homework Statement Solve the initial value problem and determine at least approximately where the solution is valid (2x-y) + (2y-x)y' = 0 y(1) = 3 Homework EquationsThe Attempt at a Solution I know how to solve it, and I got the correct answer, which was: 7 = x^2 - yx + y^2 and then applying...
  17. Mark Brewer

    Exact Differential Equations for Level Curves of u(x,y) = cos(x^2-y^2)

    Homework Statement Given u(x.y), find the exact differential equation du = 0. What sort of curves are the solution curves u(x,y) = constant? (These are called the level curves of u). u = cos(x2 - y2)The Attempt at a Solution partial derivative du/dx = (-2x)sin(x2 - y2)dx partial derivative...
  18. M

    Differential equations: Circuit RL

    Homework Statement A simple electrical circuit consists of a voltage source E(t) = t*e-t volts, a resistor R = 1 and an inductor L = 1/10 H connected in series. It is assumed that I(0) = -10/81 a) The differential equation that governs the current I (t) in this circuit . b) Find the time...
  19. S

    Learning Differential equations, week 2 level material

    Homework Statement Find the general solution to the equation. Homework Equations (dy/dx) - y - e^3x=0 The Attempt at a Solution [/B] I rewrote this as dy/dx - y = e^3x This is a linear first order ODE, in the form dy/dx + P(x)y = f(x) P(x) = 1; f(x) = e^3x The integrating factor =...
  20. D

    Third Order Differential Equations (Mass-Spring-Damper-etc)

    So I'm not sure whether this should be posted in the Physics forum or not, but here goes: I'm a Junior ME student at NC State taking a Vibrations course. We've gone over the general differential equation for mass-damper-spring systems where Meq⋅x'' + Ceq⋅x' + keq⋅x = Fextermal What I was...
  21. Q

    Solution to Coupled Second Order ODE's

    Homework Statement [/B] I'm trying to 'solve' two coupled second order ODE's with the intent of putting them in state space. My specific problem is more complex and includes additional equations which are irrelevant. Essentially I can solve the problem if I know the solution to this. x1 and...
  22. D

    System of ODE - comparison with paper

    I have the following system of differential equations, for the functions ##A(r)## and ##B(r)##: ##A'-\frac{m}{r}A=(\epsilon+1)B## and ##-B' -\frac{m+1}{r}B=(\epsilon-1)A## ##m## and ##\epsilon## are constants, with ##\epsilon<1##. The functions ##A## and ##B## are the two components of a...
  23. T

    Zero-Input/Zero-State Response vs. Homogenous/Particular Solution

    I have a question regarding the solutions to linear-ordinary differential equations. I had originally learned that the solutions to such differential equations consist of a homogenous solution and particular solution. The homogenous response is due to initial conditions while the particular...
  24. VoteSaxon

    Finding the Particular Integral for d2y/dt2 + 4y = 5sin2t

    Homework Statement Well I am looking for the particular integral of: d2y/dt2 + 4y = 5sin2t The attempt at a solution As f(t) = 5sin2t, the particular integral yPI should look like: yPI = Acos2t + Bsin2t dyPI/dt = -2Asin2t + 2Bcos2t d2yPI/dt2 = -4Acos2t - 4Bsin2t Subbing into the differential...
  25. Mark Brewer

    Ordinary Differential Equations (ODE) Problem

    Homework Statement dy/dx + 2sin2pix = 0 -------Answer: y = 1/pi cos2pix + c Homework EquationsThe Attempt at a Solution I made several attempts but no success to the correct answer. The first step I made was subtracting 2sin2pix to both sides. I then used integration by parts, and this is...
  26. H

    Seek power series solutions of the given differential equation

    I know there are a number of ways to do this problem, to increment the series etc. but, would someone please be able to explain how they get the answers for this problem simply and easily ? Thanks! A screen shot is attached
  27. T

    Ordinary Differential Equations - Existence/Uniqueness Proof

    I'm having some difficulty with a problem from Boyce & DiPrima's Elementary Differential Equations and Boundary Value Problems, 9th Edition. The problem comes from Section 2.8: The Existence and Uniqueness Theorem and is part of a collection of problems intended to show that the sequence...
  28. J

    Setting up differential equations for voltage

    Let V1 be the voltage across C1 and V2 be the voltage across C2. I want to solve for V1 and V2 as a function of time. My idea was to use dV1/dt=I1/C1 and dV2/dt=I2/C2. Then using circuit rules i can express I1 and I2 as functions of V1 and V2 and substitute them into the previous diff eqs...
  29. Luterinho

    Ordinary Differential Equations by Tenenbaum and Pollard

    I am having a hard time understanding the conditions that set a plane to be called a region. According to definition 2.68, a set in the plane is called a region if it satisfies the following two conditions (p. 14): 1. "Each point of the set is the center of a circle whose entire interior...
  30. kostoglotov

    Will linear algebra help me in more advanced diff eqs study?

    I just finished up Stewart's Calculus Textbook, and the last section was on solving 2nd Order Non-Homogeneous Diff Eqs using power series. I've looked through Paul's Calculus page in the Differential Sections, and can see that there is still a lot more beyond Stewart's that I'd like to study...
  31. kostoglotov

    Series Soln to a Diff Eqn: can't understand one of the steps

    Homework Statement solve y" - 2xy' + y = 0 Homework EquationsThe Attempt at a Solution in the worked example, the book gets from here: \sum\limits_{n=0}^{\infty} (n+1)(n+2)c_{n+2}x^n - \sum\limits_{n=1}^{\infty}2nc_nx^n + \sum\limits_{n=0}^{\infty}c_nx^n = 0 to here...
  32. K

    Differential equations time evolution

    Homework Statement Homework EquationsThe Attempt at a Solution Any help would be appreciated
  33. kostoglotov

    Trial solution, undetermined coefficients, 2nd order non-homogeneous equation

    Homework Statement Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients. For: y'' + 2y' + 10y = x^2e^{-x}\cos{3x} There's a modification performed and I'm not 100% confident as to why. Homework EquationsThe Attempt at a Solution The...
  34. L

    First order differential equations and constants

    Homework Statement ##y' = \frac{cos x}{sin y}## ##y' = \frac{6x^2}{y(1 + x^3)}## Homework EquationsThe Attempt at a Solution So I was working through some textbook problems and there's something about the solutions of the above equations I don't quite understand. The first one: ##\int sin...
  35. MexChemE

    Non-dimensionalization of the energy balance

    Good evening people of PF! I have recently encountered a problem from Himmelblau's Basic Principles and Calculations in Chem. E. which asks to set up an energy balance for a tank, and then non-dimensionalize the differential equation before solving it. It's not the most complex task, but it's...
  36. Lagraaaange

    Take Partial Differential Equations? Senior

    Textbook by Asmar. Would this class help me a lot for grad courses, like Jackson Electrodynamics or Sakurai Quantum? Debating to just finish up my upper levels and get As
  37. L

    Intro Math Mastering Differential Equations

    During the summer, I plan on learning differential equations (ODE's and PDE's) from bottom to top, but I am unable to choose books due to a great variety present. Can you suggest books for me to read in the following order (you can add as many books in each section if you like);Ordinary...
  38. kostoglotov

    Another Diff Eq's one: am I wrong?

    Homework Statement Sorry to be bringing these in quick succession, but this one really has me perplexed. Is it possible that both the solution manual and I have two different but valid answers? I don't want to just go assuming that I'm right...but I think by subsuming certain parts of the...
  39. kostoglotov

    Diff Eqs: can't get a reasonable answer

    Homework Statement Imgur link: http://i.imgur.com/C1epJzI.png My issues arise in part (b) but cause me to doubt my solution to part (a) as well. Homework EquationsThe Attempt at a Solution \frac{dx}{dt} = k(a-x)(b-x) \int \frac{1}{(a-x)(b-x)} dx = k \int dt Use partial fraction...
  40. Destroxia

    Undetermined Coefficients, Differential Equations

    Homework Statement y'' + y =3*sin(2t) +t*cos(2t) Okay, so I have found the complimentary solution, and the first partial solution as listed in my work below. My problem is the work on the second partial solution. I have got all the derivatives plugged into the differential equation, my...
  41. M

    MHB Differential equations - Decidability and Complexity

    Hey! :o Is someone familiar with the following? We have linear differential equations with polynomial coefficients depending on x. $a_n(x)y^{(n)}+ \dots a_1(x)y^{(1)}+a_0(x)y^{(0)}=b(x)$ There are problems like if there are solutions, if the solutions are linear independent and so on and...
  42. T

    MHB Differential Equations of Calculus Problems- Calc I course

    So there are these 4 differential equations problems that are ruining my day. I can't find any videos or notes for differential equations of calculus which is for some reason in my Calc 1 class. They seem kind of simple but I just can't seem to get started... any help is appreciated! 1b) When...
  43. Destroxia

    Solving Differential Equations: Challenges & Solutions

    Homework Statement Solve the Differential Equation:[/B] When I take the partial derivative of each of these equations, I do indeed get that it is exact... However, when I do it the way my professor wants me to do it, I don't get the same result. He told us to multiply through by the common...
  44. Destroxia

    Differential Equations, Separable, Simplification of answer

    Homework Statement I believe I have solved this differential equation, yet do not know how the book arrived at it's answer... Solve the differential equation in its explicit solution form. The answer the book gives is... Homework Equations Separable Differential Equation The Attempt...
  45. Destroxia

    Differential Equations, Separable, Explicit Solution

    Homework Statement Solve the differential equation, explicitly. dy/dx = (2x)/(1+2y) The answer given by the book is... -1/2 + 1/2sqrt(2x - 2x^2 +4) Homework Equations Process for solving separable differential equations The Attempt at a Solution dy/dx = (2x)/(1+2y) (1 + 2y)*dy = 2x*dx...
  46. H

    Separable differential equations

    Homework Statement [/B]Homework Equations The Attempt at a Solution I've highlighted two equations on the screenshot. How did it proceed from the first to the second? I'm actually confused with the absolute values. What is the idea behind getting rid of the first absolute value(1-5v^2) while...
  47. S

    Differential equations [arctan(x) to tan(x)]

    Homework Statement dx/dt = (1+x2)et ; x(0) = 1 1/1+x2 dx/dt = et ∫ 1/1+x2 (dx/dt) * dt = ∫ et dt ∫ 1/(1+x2) dx = ∫ et dt arctan(x) = et + c so x(t) = tan (et+c) Homework EquationsThe Attempt at a Solution dx/dt = (1+x2)et ; x(0) = 1 1/1+x2 dx/dt = et ∫ 1/1+x2 (dx/dt) * dt = ∫ et dt ∫...
  48. L

    Frequency of a wire/spring system?

    Hi, I have a wire that is fixed on one end and is under 2 lbs (approx) of tension on the other end due to a spring. Using a fixed/fixed closed form solution I found the first mode to be 73.6 Hz. In a vibration test, we found the first mode to be 110 Hz. Does this seem reasonable and where do...
  49. H

    System of nonlinear differential equations

    Hello I have a system of differntial equations: dx/ds = sin(p) dy/ds=cos(p) dp/ds = k dk/ds = -1/EI(s)*(k*dEI/ds+f*sin(p)) x(0)=y(0)=p(0)=p(L)-pl = 0 These are nonlinear differential equations. I should use some sort of nonlinear finite difference. But I do struggle to setting up the finite...
  50. J

    Doubt about convergence test on differential equations

    I will try to explain this with an analogy. Let's have this equation: x^2 =9 And let's assume I don't know algebraic methods to solve it, so I create a list using excel with different values. And I see that if I put x=4 it doesn't work, if I put x=5 it is even worse and so on. But If I put...
Back
Top