What is Differential equation: 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. A

    Differential equation 5y4y' = x2y' + 2xy

    Homework Statement Find the general solution of: 5y4y' = x2y' + 2xy 2. The attempt at a solution Well I've so far tried to simplify by making the equation really: (5y4-x2)y' - 2xy = 0 Now this will let us use exact equations such as: N(x,y)= 5y4-x2 and M(x,y)= -2xy...
  2. C

    Superficially simple vector differential equation problem

    Hi, I have the following vector differential equation (numerator layout derivatives): \frac{\partial e(v)}{\partial v}=\frac{1}{\beta} \frac{\partial w(v)}{\partial v} \Gamma^{-1} where both ##e(v)## and ##w(v)## are scalar functions of the vector ##v##, and where ##\Gamma## is a...
  3. N

    Need help with partial differential equation

    Homework Statement Given that z = √3x/y show that: Homework Equations ∂2z/∂x∂y = ∂2z/∂y∂x The Attempt at a Solution
  4. E

    Find the integrating factor u(x) for this exact differential equation

    Homework Statement (3x^2y+2xy+y^3)dx+(x^2+y^2)dy=0
  5. P

    Differential equation parametrisation with integrating factor

    Homework Statement Use parametrisation first, derive the equation including y and p = \frac{dy}{dx} and use the integrating factor method to reduce it to an exact equation. Leave the solution in implicit parametric form. (y')^{3} + y^{2} = xyy'The Attempt at a Solution I'm really lost at...
  6. L

    Differential equation: word problem

    1. In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For the change of δ-glucono lactone into gluconic acid, for example, dy/dt= -.6y when t is measured in hours. If there are 90 grams of δ-glucono lactone present...
  7. L

    Differential equation: dy/dx = e^(8x-3y), y(0)=16

    1. The next differential equation I'm working on is this: dy/dx= e8x - 3y Alright, I thought to cancel out the e, we could take the ln of both sides...? ln(dy/dx) = 8x-3y Is this right so far? It doesn't seem like it's right because... well, how can you have an ln(dy/dx)?? Would...
  8. L

    Differential equation: 4sqrt(xy)dy/dx=1, y(1)=1

    The differential equation I'm working on is: 4(√(xy))dy/dx=1, y(1)=1 (√(xy)dy/dy)2 = (1/4)2 ((xy)dy/dx)*dx = (1/8)*dx (xy)dy = (1/8)dx (y)dy = (1/(8x))dx ...So I think this is right so far. Now I'm going to take the integral of both sides. ∫(y)dy = ∫(1/(8x))dx...
  9. T

    Finding the maximum value of x in a differential equation.

    Homework Statement I want to find the maximum value of x given that the differential equation relating x is: x' = 0.7548 - 599.49x^6 -2.4038x^2 - 0.12236x^1.5 Where x = 0 at t = 0. I have to do this with MATLAB using Euler's Method and here is my code: x = 0 a = 0 % initial t value b = 1000...
  10. N

    Which is the correct expression for the Differential Equation

    Homework Statement Which of the following is a correct expression for ∫1t(s-1)y'(s) ds I know the answer is: a) 1/2(t-1)2y(t) b) 1/2(t-1)2y'(t)+ (t-1)(y(t)-y(1)) c) (t-1)(y(t)-y(1)) d) (t-1)y(t)-∫y(s) ds Homework Equations The Attempt at a Solution I know the...
  11. N

    Which of the Following is a solution to the differential Equation

    Homework Statement Which fot he following is a solution to the differential equation: t2y" - 2ty' + 2y = 0 a)e-2tt3 b)et c)t d) t4 I have attached an image of the question Homework Equations The Attempt at a Solution I tried answer d first: y = t4 y' = 4t3...
  12. N

    The solution to the Differential Equation

    Homework Statement The Initial value problem: y' - (3/t)y = 0 y(1) = -10 Has the solution: I have attached an image of the question Homework Equations The Attempt at a Solution Firstly I found the integrating factor: u(t) = e∫-3/t dt u(t) = -3/t I plugged...
  13. T

    Differential Equation Help: As t approaches 0, y approaches

    Homework Statement http://edugen.wileyplus.com/edugen/shared/assignment/test/session.quest1886032entrance1_N10020.mml?size=14&rnd=1360201586591 (b) Solve the initial value problem and find the critical value a0 exactly. y = ? a0 = ? (c) Describe the behavior of the solution...
  14. G

    Green function Helmholtz differential equation, monodimensional limit

    The solution of the problem \left(\nabla^2 + k^2 \right)\psi(\mathbf{r})=f(\mathbf{r}) is, using green function \psi(\mathbf{r})=-\int G(\mathbf{r},\mathbf{r}_1) f(\mathbf{r}) where for the tridimensional case the Green function is...
  15. I

    Solve the integral of differential equation

    Homework Statement Solve the integral of differential equation ∫^{t}_{0} y(\tau) d\tau -\acute{y} (t) = t for t≥0 with y(0)=4 The Attempt at a Solution take laplace both sides \frac{Y}{s} - sY + 4 = \frac{1}{s^{2}} Y \frac{1 - s^{2}}{s} =\frac{1}{s^{2}} - 4 Y...
  16. S

    Separable differential equation

    Homework Statement I am asked to find a singular solution of the D.E. dy/dx = (xy+2y-x-2)/(xy-3y+x-3). I am first solving to find the general solution form of the D.E., and so far have it to: [(x+2)/(x-3)]dx = [(y+1)/(y-1)]dy From here, of course, you integrate both sides, but I am...
  17. H

    Help with Differential equation

    A hemispherical bowl of radius a has its axis vertical and is full of water. At time t=0 water starts running out of a small hole in the bottom of the bowl so that the depth of water in the bowl at t is x. The rate at which the volume of water is decreasing is proportional to x. Given that the...
  18. M

    Eliminating Arbitrary Constants: Solving the Differential Equation

    Homework Statement From the differential equation by eliminating the arbitrary constant from the equation (y - b ) ^2 = 4 (X-a ) http://www7.0zz0.com/2013/02/02/21/747681463.png
  19. Fernando Revilla

    MHB Norwin Dole's question at Yahoo Answers about a differential equation

    Here is the question: Here is a link to the question: How to obtain the general solution of ye^(2x) dx = (4+e^(2x))dy? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  20. P

    Solving time dependent differential equation and plotting

    Hi i have the differential equation \frac{d^{2}}{dt^{2}}X(t) +(A+B\frac{sin^{2}(mt)}{mt})X(t) I have tried by hnd to solve this and am getting knowhere does anyone know how to solve it and then plot X against t (where the constants A, B and m will be arbitrarily added), possibly using maple? I...
  21. Fernando Revilla

    MHB Non homogeneous linear differential equation

    I quote an unsolved question posted in MHF (November 7th, 2012) by user NumberMunhcer.
  22. T

    Is the Black-Scholes equation a differential equation?

    Hi everyone, first post. To anyone who has had experience with the background of the Black-Scholes equation used in finance to price options based on underlying assets (wiki here), I have just one simple question to ask regarding a research paper I must write. Is this equation a stochastic...
  23. P

    Unusual Solution to Riccati Differential Equation with ln x Particular Solution

    Homework Statement This is a Riccatti Equation which my answer is very weird after i solved it.. dy/dx+2y^2=(y/x lnx)+2(ln x)^2 where y=ln x is a particular solution Homework Equations The Attempt at a Solution i first let y=ln x+ 1/w and dy/dx=1/x-1/w^2 and substitute y and...
  24. E

    Differential equation dv/dt = 9.8 - v/5, v(0) = 0

    A falling object satisfies the initial value problem: dv/dt = 9.8 - v/5, v(0) = 0 1.Find the time that must elapse for the object to reach 98% of its limiting velocity. answer: t = 19.56, and for completeness, v = -49e-t/5 + 49 2.How far does the object fall in the time found in part a...
  25. S

    Differential Equation Involving Newtons Law of Motion

    Homework Statement mv'=-gm-kv Find the position function using the initial coniditions of t=0 for all Constants Homework Equations Reverse product rule The Attempt at a Solution My attempt is on my white board. Its attached as a picture.
  26. D

    Engineering RLC circuit - Differential equation - Stationary current calculation

    Homework Statement I'm decent with differential equations for RLC circuits, and I KNOW the stationary current will be zero, but I need help to work it out mathematically, because my maths gets me -2.4A (...) L: 0.05H C: 0.04F R: 3 Ω U (source) = 1 V I've got the transient solution: c1e-10t +...
  27. B

    Brachistochrone Differential Equation

    The first part of this problem asks me to solve the following for y' : \left( 1 + {y'}^2 \right)y = k^2 So I have: 1 + {y'}^2 = \frac{k^2}{y} {y'}^2 = \frac{k^2}{y} - 1 y' = \sqrt{{\frac{k^2}{y} - 1}} Then I am asked to show that if I introduce the following: y = k^2 \sin^2t Then...
  28. P

    Non-exact differential equation

    the problem is (x+x^4)dy+y(y^3-x^3)dx=0 well I know that this is not a separable equation, homogenous equation or an exact equation...so i try to solve it by treating it as a non exact DE by finding out the integrating factor...but the both IF come out in term of x and y which involve 2...
  29. D

    Help with this separable differential equation

    The problem is dx/dt = (x+9)^2. This is separable so I made it dx / (x+9)^2 = dt. The only method I can think of using for something like this is partial fraction, but I can't get it to work with A/(x+9) + B/(x+9). Can anyone find a method that works?
  30. R

    Differential Equation First Order Linear

    Homework Statement Solve the following differential equation: y*e^(x^2)*dy/dx=x+xy Homework Equations y'+P(x)*y=Q(x) The Attempt at a Solution I tried to modify the equation to match the first order linear one, and I got: e^(x^2)*dy/dx=x/y+x (divided everything by y), but...
  31. U

    Solving a First Order Differential Equation with Initial Conditions.

    Homework Statement Solve the initial value problem: t(dy/dt)+8y=t^3 where t>0 and y(1)=0 Homework Equations None? The Attempt at a Solution It's a linear equation, so rearranged to dy/dt+8y/t=t^2. Took the integrating factor e^(∫8/tdt)=t^8 and multiplied through...
  32. I

    First order differential equation question

    The problem is : dy/dx=(x(x^2+1))/4y^3 when y(0)=-1/√2 This is my work so far: ∫4y^3dy=∫x(x^2+1)dx (y^4)/2=((x^2+1)^2)/2+c The answer from the textbook is y=-(√(x^2+2)/2) As you can see, my work will never equal the textbook answer when you put it in the y= stuff form. What did I do wrong?
  33. A

    What is the Correct Approach for Solving This Differential Equation?

    Homework Statement y = 2xy' + y(y')2 ; y2 = c1(x + c1/4) Homework Equations So far I've gotten the second equation to be: y = (c1x + c12/4)1/2 I was then going to take the derivative of that equation and plug them into the first equation after setting it to zero. Is that the...
  34. C

    Boundary Conditions and Solutions to Differential Equations

    Homework Statement Show that f(x) = A exp(σx) + B exp(-σx) is a solution to the following differential equation: f''(x) = (σ^2)f(x) where A, B, and σ are constants. What if a boundary condition is included that f(-∞) = 0? Homework Equations differential equation: f''(x) =...
  35. W

    Matrix differential equation for rectangular matrix

    Given a matrix differential equation (system of equations?) of the form: \textbf{X}^{\prime}(t) = \textbf{AX}(t) (where X is a complex matrix, t is real scalar and A is always a square and normal real matrix) I am able to find (e.g. here) that a general solution for square \textbf{X} is...
  36. marellasunny

    Number of parameters for a nonlinear differential equation

    Take for example a system \frac{dx_i}{dt}=(x_i,t,a,b,...) i-number of state equations. What would be the maximum number of parameters permitted for this system of non-linear differential equations? Is it finally determined by the solution space?Is there a criteria for number of...
  37. C

    I'm starting to learn about differential equation

    Homework Statement Verify that the differential equation, {\frac{dy}{dx}} = 15 - {\frac{2y}{81+3x}} has the general solution y(x) = 3(81+3x) + C(81+3x)^{-2/3} 2. The attempt at a solution I've just learned about differential equations, so I'm probably missing something very...
  38. I

    Separable Differential Equation Question

    Homework Statement dy/dx = (y^2 - 1)/ (x^2 - 1) with initial condition y(2) = 2 Why is y = 1 and/or y= -1 not solutions? Homework Equations The Attempt at a Solution I am actually able to solve this differential equation but when I separate the equation according to x and y...
  39. M

    Numerically Solving a Second Order Differential Equation Using C

    Homework Statement The Second Order Differential Equation is: x''-u(b^2 + x^2)x'+x=0 Initial Conditions are: x(0)=1 x'(0)=0 It is to be numerically solved for 0<=t<=500. The specific numerical method to be used isn't specified, but it must be programmed into c. As a means to check the...
  40. H

    How are partial differential equations used to model physical systems?

    Plese give me silminer simple example or anther example on this case or explein the steps
  41. S

    Non-exact differential equation

    I try to show, that equation \frac{-y}{ x^{2}+y ^{2} } + \frac{x}{ x^{2}+y ^{2}}y'=0 is not exact in \mathbb{R^{2}} \setminus \{(0,0)\}. It's obvious that I have to use the fact, that the set is not simply connected, but I don't know how to do it.
  42. J

    Can This Differential Equation Be Simplified or Solved Numerically?

    I have the following equation \frac{\partial}{\partial y}\left(y\frac{dm}{dx}+m\frac{dy}{dx}\right)-\frac{dm}{dx}=0 where m is a function of y (say m=f\left(y\right)) and y is a function of x (say y=g\left(x\right)). Are there any conditions under which \frac{dm}{dx} becomes identically...
  43. A

    Rayleigh's differential equation

    In Rayleigh's DE : http://www.wolframalpha.com/input/?i=rayleigh+differential+equation What does mu stand for?
  44. J

    How to integrate this partial differential equation

    I have the following equation \frac{\partial}{\partial y}\left(m\frac{dy}{dx}\right)=0 where y is a function of x and m is a function of y. If I integrate this equation first with respect to y should I get a function of x as the constant of integration (say C\left(x\right)) or it is just...
  45. B

    Differential equation with only the trivial solution

    Homework Statement Find a differential equation with its only (complex-valued) solution being y=0Homework Equations The Attempt at a Solution I believe that there is no DE having only y=0 as its solution, but frankly I am not sure if this is the case. I would like to know whether or not this is...
  46. J

    Nonlinear second order differential equation

    What is the solution of the follwoing differential equation \frac{\partial^{2}y}{\partial x^{2}}-ay^{-1}\frac{dy}{dx}=0 where a is a constant.
  47. J

    Differential equation question

    What is the logic behind equating differential equations to zero? For example the equation y''-5y'+6y=0 Because it can just as easily be written y''-5y'=-6y I am interested in the meaning of why if we sum y''+(-5y')+6y equals zero. What is the relationship of its second derivative, first...
  48. J

    Differential equation selection and linear transformations

    This may be vague, so I apologize. I am interested in applied mathematics, so my question is about the process a scientist or engineer uses to determine what differential equation to use for a non-linear process. I am not familiar enough with describing non-linear processes to be able to...
  49. E

    Does this differential equation have a closed form?

    I was busy doodling and basically ended up constructing this differential equation: p'(t)=c(t)p(t)-c(t-T)p(t-T) Obviously I've dealt with eq's like p'(t)=c(t)p(t) but I'm getting stuck because of the second term. Does this differential equation even have a closed form? Thanks.
  50. D

    Partial differential equation, characteristic equations.

    Homework Statement Given the initial value problem: \frac{(u)}{(1-e^-(2x))}u_{x}+ \frac{\sqrt{t}}{u}u_{t}=1, with x, t, u > 0 Subject to condition u(x,1)=e^{-x} Homework Equations a) Classify given partial differential equation. b) Write the characteristic equations. By...
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