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. D

    Linear first order differential equation with unknown function

    I was wondering if there is a way to get specific numerical values for the following differential equation: f'(x)+ \frac{1}{x-20}\cdot f(x)=\frac{1}{x-20}\cdot g(x) I have numerical values for g(x) for about 10 different x values. I need to find f(x) numerically for those same values...
  2. F

    Differential equation for projectile

    I want to model a Differential Equation for a projectile motion under 2 forces (gravity and wind) So, what I have now is an algorithm that simulate the parametric motion (2d) of the project under those 2 forces (given a P position of the projectile with velocity V under a vector of forces F (or...
  3. R

    Find the solution to this Differential Equation

    Homework Statement Find the solution to ## \frac{dy}{dx} = yLn(y + 1) ## Homework Equations ## \frac{dy}{dx} = yLn(y + 1) ## The Attempt at a Solution ## \frac{dy}{dx} = yLn(y + 1)## ## \frac{dy}{yLn(y + 1)} = dx ## but then i can't integrate, any help?
  4. V

    Power series solution to differential equation

    Homework Statement Find the power series solution of the differential equation y''-\frac{2}{(1-x)^2}y=0 around the point ##x=0##. Homework Equations y=\sum_{n=0}^\infty{}c_nx^n y'=\sum_{n=0}^\infty{}c_{n+1}(n+1)x^n y''=\sum_{n=0}^\infty{}c_{n+2}(n+2)(n+1)x^n The Attempt...
  5. S

    Yet another first order differential equation

    Homework Statement okey, so i got stuck at another step in the way of solving de's.I've been studying DE of this form: y' + P(x)y = Q(x) Homework Equations The Attempt at a Solution So, first we solve y' + P(x)y=0 for y. \frac{dy}{y} = -P(x)dx , we integrate this and get...
  6. S

    First order differential equation

    Homework Statement I'm starting college this autumn(physics) and I started learning some calculus on my own, basic stuff like first order differential equation and so on.Recently i stumbled on something that i don t understand.I was reading the course and re-solving the given examples for...
  7. E

    Homogenous differential equation

    My book says: "Differential equations of the form $$\frac{\mathrm{d} y}{\mathrm{d} x}=f(x,y)$$, where $$f(x,y)$$ is homogenous function (function is homogenous if $$f(tx,ty)=t^k f(x,y)$$) can be written in form $$\frac{\mathrm{d} y}{\mathrm{d} x}=F(\frac{y}{x})$$ and transformed to differential...
  8. C

    MHB Stochastic Differential Equation using Ito's Lemma

    I am new to SDE, and especially Ito's Lemma. I have a question that I simply cannot answer. It attached. Can someone please help?
  9. E

    Trouble with a separable differential equation

    I have this equation: dy/dx = 1-y^2, so then dy/(1-y^2) = dx, so ∫dy/(1-y^2) = dx ---> ∫(A/(1+y) + B/(1-y))dy = x + C. I rewrite it again: (A - Ay + B + By)/(1+y)(1-y) = 1/(1+y)(1-y) so I get A+B = 1, and B-A = 0, so B = A, and therefore 2A = 1. so A & B = 1/2. So 1/2∫(dy/(1+y) +...
  10. Greg Bernhardt

    What is a linear ordinary differential equation

    Definition/Summary An nth-order linear ordinary differential equation (ODE) is a differential equation of the form \sum_{i=0}^n a_i(x)y^{(i)}(x)\ =\ b(x) where y^{(i)}(x) denotes the ith derivative of y with respect to x. The difference between any two solutions is a solution of the...
  11. johann1301

    Why Can't I Use dy Instead of dx in Solving This Differential Equation?

    Homework Statement a) Write (x21)y'+2xy as the derivative of a product b) Solve (x21)y'+2xy=e-x The Attempt at a Solution a) I use the product rule backwards and get ((x2+1)y)' b) I exploit what i just found out... (x21)y'+2xy=((x2+1)y)' and get... e-x=((x2+1)y)'...
  12. johann1301

    Is this a differential equation?

    My textbook says that: "A differential equation contains both the function and the derivative of the function" and at the same time claims that y'=3x2 is a differential equation. How can this be? The original function isn't part of the equation in this case?
  13. johann1301

    Problem with a differential equation

    Homework Statement Show that a solution to y'=y(6-y) has a an inflection point at y=3. The Attempt at a Solution If y has an inflection point, then y''=0. I know that y'=y(6-y), and therefore i know that y''=(y(6-y))'=(6y-y2)'=6-2y So, if y''=0, and y''=6-2y then 0=6-2y => y=3. Solved. But...
  14. S

    Solution of differential equation with Dirac Delta

    Is it possible to solve a differential equation of the following form? $$\partial_x^2y + \delta(x) \partial_x y + y= 0$$ where ##\delta(x)## is the dirac delta function. I need the solution for periodic boundary conditions from ##-\pi## to ##\pi##. I've realized that I can solve this for some...
  15. blintaro

    A Simple(?) Differential Equation

    This is question 1.4 of Chapter 8 of Mary L. Boas's Mathematical Methods in the Physical Sciences, Edition 2. I'm using it as a substitute for my ordinary differential equations class since my textbook has apparently been lost somewhere in the mail. Homework Statement Find the distance...
  16. C

    Non linear 2nd order differential equation

    please provide step by step method to solve this 2nd order non linear differential equation: attached with this thread. take FUCOS(Ѡt) and FUsin(Ѡt) as zero.
  17. A

    Partial Differential Equation with initial conditions

    Hello! This is my first post to this excellent forum! I would like some help with this exercise: u_{xx} (x,y) + u_{yy} (x,y) = 0, with 0 < x < 2 \pi , 0 < y < 4 \pi u_x (0,y) = 0, \, u_x(2 \pi, y) = 0, \, 0< y < 4 \pi u(x,0) = a \cos(2x), \, u(x, 4 \pi) = a \cos^3(x), \, 0<x<2\pi...
  18. S

    Need help solving a differential equation for orbit.

    I want to be able to map the position of a planet given initial position, velocity, and acceleration. I know the equation for Gravitational force (Newtonian) is: F=-GMm/r^2 Using Newtons second law this gives: m(d^2x/dt^2)=-GMm/x^2 Then simple Algebra yields: (d^2x/dt^2)+GM/x^2=0 I...
  19. Portal.Leaf

    Obtain the differential equation of the family of plane curves

    Homework Statement Obtain the differential equation of the family of plane curves described: Circles tangent to the x-axis. Homework Equations (x-h)^2 + (y-k)^2 = r^2 The Attempt at a Solution I tried to answer this question using the same way I did on a problem very similar to this...
  20. H

    Desperately seeking help with solving a differential equation

    Hi, I desperately need help to solve the following differential equation for buckling of a beam with a uniform axially applied force and a point force: ∂y(x)2/∂x2+(P+Q.x).y(x)=0 Where P and Q are constants. P is known and Q is the critical axial uniform force (N/mm) that will cause...
  21. S

    Solve partial differential equation

    Homework Statement Solve PDE ##\frac{\partial u}{\partial t}-k\frac{\partial^2 u}{\partial x^2}=0## for ##0\leq x \leq \pi ## and ##t\geq 0##. Also ##\frac{\partial u}{\partial x}(0,t)=\frac{\partial u}{\partial t}(\pi ,t)=0## and ##u(x,0)=sin(x)##.Homework Equations The Attempt at a Solution...
  22. S

    MHB Confirming what I learned, a follow up differential equation

    With some help in this other thread http://mathhelpboards.com/calculus-10/differiental-equation-question-particular-solutions-10864.html I was able to see what I was doing wrong. Now I'm going to apply it to a different problem and see if I'm doing it right. dy/dx+(3/x)y=-16sin x^4, y(1)=1...
  23. G

    Differential equation to describe beam width

    I have two independent variables (rotation in degrees, and distance from best focus distance in mm) and one dependent variable (1/e^2 width squared in mm^2). How would I go about creating either a multivariable function or a differential equation to describe the data? I've attached the raw data...
  24. S

    MHB Problem with linear differential equation

    Still learning the formatting commands, sorry! I'm aware of the $(dy/dx) + P(x)y=Q(x)$ formula, as well as the $e^{\int P(x) dx}$ formula needed to get the "I" factor. Here's the equation. $$(dy/dx)+(2/x)y=3x-5$$ The "$P(x)$" would be $(2/x), \int 2/x\ dx = 2 \ln(x), e^{2 \ln(x)} = x^2$, so...
  25. M

    A differential equation question

    Verify that given function is a solution. y'' - 2y' + 2y = 0 , y=e^x(Acos x + Bsin x) First I take derivative of y which is y+e^x(-Asin x + B cos x) then I asign e^x(-Asin x + B cos x) to y'-y. Then I take derivative of y' which is y'+(y'-y)-y which equals 2y'-2y=y'' then I use y'' as...
  26. M

    A differential equation question

    There is differential equation with initial condition perplexing me. y'+ y = 1, y = ce^-x + 1 , y = 2.5 when x = 0 First I take derivative of y which is -ce^-x then I sum it up with y which is -ce^-x+ce^-x + 1 equals 1 which is in harmony with y' + y = 1 but it seems that this is...
  27. V

    Shoot the moon - differential equation for motion in earth - moon syst

    Homework Statement We have given coordinates on the Earth from where we are shooting to the Moon (bullet has really small mass). The Moon orbit and therefore Moon position in time t is known. The task is to compute the initial velocity vector (the angle and velocity of the bullet), so the...
  28. E

    Help with a differential equation

    Homework Statement Hello I am new to this forum, but I hope I can get help with a problem I haven't been able to figure out what to do with. info: we have a one dimensional equation -d/dx [a(x) du/dx] = p(x) where we seek a solution u(x) where x is within [0,1] , that satisfies...
  29. C

    Differential Equation Question

    I have a differential equation y'' + y' -2y = 3e-2x + 5cosx y = yc + yp I found yc = Ae-2x + Bex for A, B arb. Const. Then when selecting a trial function to find the particular integral, yp I came up with: yp = ae-2x + bcosx + csinx However the correct trial function...
  30. C

    Choosing a Trial Function for Differential Equation Homework

    Homework Statement I have a differential equation: \ddot{x} -2\dot{x} + 5x= 10 + 13cos(3t)Homework Equations x(t) = xc + xp where xc is the Complementary Function and xp is the Particular Integral.The Attempt at a Solution I have formed and solved the auxiliary equation: m^{2} - 2m + 5 = 0...
  31. I

    How Do You Calculate and Sketch the Light Cone in a 2D Space-Time Geometry?

    Homework Statement We are given a 2 dimensional space time line element and we want to calculate the light cone at a point (x,y) Homework Equations ds^2=x(dy)^2-2(dy)(dx) The Attempt at a Solution For a light cone, ds^2=0 so x(dy)^2-2(dy)(dx)=0 now what?
  32. J

    Separation of variables for solutions of partial differential equation

    Why is it assumed that the method of separation of variables works when the boundary conditions of some boundary valued problem are homogeneous? What is the reasoning behind it?
  33. K

    Laplace Transform solution for 2nd order differential equation

    Homework Statement d^2x/dt^2 - 3 dx/dt + 2x= 2e^3t give that at t=0, x=5, and dx/dt=7 Homework Equations i can't figure out how to derive the values of A, B, and C from the attempted equation solution. please help me out here. thanks The Attempt at a Solution
  34. E

    What is the solution of this differential equation?

    how can we solve this differential equation? (ax+y)∂f/∂x + (ay+x)∂f/∂y =0 with this condition: if a=0 then f(x,y)=x^2 - y^2
  35. T

    Motion described with differential equation

    1. http://i.imgur.com/3xya7IM.jpg 3. I curently do not understand how to jump to finding an acceleration due to gravity (in those terms asked) from the differential equations
  36. D

    Differential equation substituition of new terms

    Homework Statement for this question, i sub u=xy and try to eliminate y in my working.. but how should i proceed since the term ux cannot be separated Homework Equations The Attempt at a Solution
  37. N

    Second order differential equation form

    A second order differential equation form d2y/dx2 = f(x,y,dx/dy) How do I read the language on the right hand side?
  38. D

    General solution of differential equation (express y in term of x)

    Homework Statement i got stucked here. below is the answer given. can anybody help please? Homework Equations The Attempt at a Solution
  39. L

    Solving a differential equation

    Homework Statement Solve (1+bx)y''(x)-ay(x)=0Homework Equations The Attempt at a Solution I'm used to solving homogeneous linear ODE's where you form a characteristic equation and solve that way, here there is the function of x (1+bx) so how does that change things?
  40. L

    Separable differential equation

    Homework Statement \frac{du}{dt} = e^{5u + 7t} Solve the separable differential equation for u: Use the following initial condition: u(0) = 6. The Attempt at a Solution I tried to take the natural log of each side but now I'm stuck. How can I separate the equation when both the u...
  41. F

    2nd Order Differential Equation with Improved Euler Method (Heun's)

    Homework Statement I would like to solve a 2nd Order Differential Equation using the Improved Euler Method. The 2nd ODE is a Mass-Spring-Damper equation. I tried coming up with an solution for the Improved Euler Method, but not entirely sure. Can you help me and have a look if this is correct...
  42. B

    Solve differential equation using power series

    Homework Statement Solve ##y^{''}+zy=0## where ##y(0)=0## and ##y^{'}(0)=1##Homework Equations ##y(z)=z^r\sum _{k=0}^{\infty } C_kz^k## The Attempt at a Solution Well firstly: ##r(r-1)+p_0r+q_0=0## where obviously ##p_0=q_0=0## so ##r_1=0## and ##r_2=1##. In general ##y(z)=\sum...
  43. AntSC

    Ordinary Differential Equation - Comparing 2 Solutions

    Homework Statement {g}'\left ( s \right )+\mu g\left ( s \right )={f}'\left ( -s \right )+\mu f\left ( -s \right ) Integrate up to get g\left ( s \right )=-f\left ( -s \right )+2\mu e^{-\mu s}\int_{-s}^{\infty }e^{-\mu {s}'}f\left ( {s}' \right )d{s}' Homework Equations As above...
  44. Matterwave

    Functional differential equation

    Homework Statement Solve: $$\frac{\delta F[f]}{\delta f(x)}=b(x)f(x)^2F[f]$$ For b(x) a fixed smooth function. Homework Equations $$\left.\frac{dF[f+\tau h]}{d\tau}\right|_{\tau=0}\equiv \int\frac{\delta F[f]}{\delta f(x)}h(x)dx$$ The Attempt at a Solution This isn't a homework problem...
  45. M

    Differential equation x^2*y'=y^2

    Hello, I have trouble solving the following differential equation. I am trying to learn how to solve that form of DEs. The DE is: x2*dy/dx = y2 There are no initial-value problem, but the solution should be given such that y is defined for all x. The most important for me is to...
  46. H

    Y''=-(t^2)y differential equation

    Hello I was recently working on a problem where I had to solve the differential equation in the title ( where y is a function of t), I found an exact series solution through peturbation theory in which a pattern emerged between successive orders. However, the series solution is not very useful...
  47. R

    MHB Non-dimensional differential equation 2

    Consider non-dimensional equation for the height at the highest point is given by \begin{equation} h(\mu)= \frac{1}{\mu}- \frac{1}{\mu^2} \log_e(1+\mu) \end{equation} $0<\mu\ll 1.$ Determine to $O(\mu)$, the (non-dimensional) time for the body to travel from the highest point to the ground, and...
  48. W

    Non-dimensional differential equation

    A body of constant mass is thrown vertically upwards from the ground. It can be shown that the appropriate non-dimensional differential equation for the height x(t;u), reached at time t\geq0 is given by \begin{equation} \frac{d^2x}{dt^2} = -1-\mu (\frac{dx}{dt}) \end{equation} with...
  49. R

    MHB Non-dimensional differential equation 1

    A body of constant mass is thrown vertically upwards from the ground. It can be shown that the appropriate non-dimensional differential equation for the height $x(t;u)$, reached at time $t\geq0$ is given by \begin{equation} \frac{d^2x}{dt^2} = -1-\mu (\frac{dx}{dt}) \end{equation} with...
  50. W

    Differential equation for concentration of gas in the atmosphere

    Homework Statement Hi everyone, I'm currently studying an online course on climate science and am a bit overwhelmed by the calculus. I have studied calculus to second year of college but that was a while ago and I'm very rusty. A few weeks ago I was a question to find the how long it would...
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