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. B

    What Is the Correct Method to Solve This Differential Equation?

    Homework Statement y3(dy/dx) = (y4 + 1)cosx 2. The attempt at a solution I solved for the homogeneous equation which is y = Ce-sinx Where C is some constant for the particular solution I tried Asinx + Bcosx where A and B are constants but when subbing in it's gets very messy. How should...
  2. MexChemE

    Unsteady state mass balance -- Draining of a tank

    Homework Statement A cylindrical tank, with a diameter Db, open to the atmosphere, is drained through an orifice in the bottom of the tank with a diameter Do. The speed of the fluid flowing through the orifice is given by v = \sqrt{2gh}, where h is the height of the liquid measured from the...
  3. R

    Trying to solve a rather difficult differential equation

    Homework Statement Consider a system composed of two species X and Y with fractional populations x and y, respectively, where x+y=1. The two species interact in such a way that the differential equation for x is: \begin{equation} \frac{dx}{dt}=xyA_{0}e^{-\alpha t} \end{equation} where $A_{0}$...
  4. 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...
  5. 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.
  6. kostoglotov

    An equation for the path that the shark will swim on

    Homework Statement [/B] A shark will in the direction of the most rapidly increasing concentration of blood in water. Suppose a shark is at a point x_0,y_0 when it first detects blood in the water. Find an equation for the path that the shark will follow by setting up and solving a...
  7. 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.
  8. 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...
  9. manifold

    A numerical solution of a second order ODE

    Hello everyone; i'd like some help in this problem : i want to solve num this differential equation { y"(t)+t*cos(y)=y } by the Taylor method second order expansion. i first have to make this a first order differential equation by taking this vector Z=[y' y] then we have Z'=[y" y'] which equal...
  10. 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...
  11. 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...
  12. 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...
  13. D

    Using analog computer to solve 2nd-order diff eq

    I'm trying to build a circuit to solve the differential equation x''+2x'+x = f(t), where f(t) is a sine wave with frequency 5Hz and amplitude 0.5V. I am supposed to get a sine/cosine wave (as the diff. eq is just the same as the ove governing spring-mass forced oscillation) as solution, but...
  14. MexChemE

    Integrating factor vs. Laplace. Engineering problems

    Hello PF! We were doing mass balances on mixing tanks in one of my ChemE courses, and in one of the problems we arrived at the following DE: \frac{dC_B}{d \theta} + 0.025C_B=0.0125 e^{-0.025 \theta} Where CB is the concetration of salt in the tank and θ is time. The professor made us solve the...
  15. Alexandre

    Is this correct second order approximation?

    I have a second order differential equation of the form (theta is a function of time): \theta ''=F\left(\theta ,\theta '\right) Turning them to two first order equations I get: \begin{cases} \theta '\:=\omega \\ \omega '=F\left(\theta ,\omega \right) \end{cases} And here's the algorithm...
  16. 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...
  17. 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?
  18. 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...
  19. c_pearls

    Classical Mechanics - finding displacement with given force

    Homework Statement - The force acting on a particle m = 3kg is given by the following force equation: F = (v/9)(3 - x2), the particle begins at a position of x = 1m with a speed of v = 0 m/s at time t = 0s. Find the displacement of the particle at time t = 5 s. Homework Equations F =...
  20. END

    How do we define "linear" for single and multivariable?

    In my multivariable calculus class, we briefly went over Taylor polynomial approximations for functions of two variables. My professor said that the second degree terms include any of the following: $$x^2, y^2, xy$$ What surprised me was the fact that xy was listed as a nonlinear term. In...
  21. 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...
  22. 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...
  23. A

    Converge pointwise with full Fourier series

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  24. 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...
  25. 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...
  26. 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 +...
  27. 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.
  28. 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...
  29. Logan Land

    MHB Need help setting up Differential Equation to solve

    dy/dx = (2y-x+5)/(2x-y-4) y(1)=1
  30. G

    Solution to differential equation

    Homework Statement This is actually a problem from my physics textbook, but I think it's mostly a mathematical problem, which is why I post it here: Show that the Langevin equation 1: \frac{dv}{dt}=-\gamma v+\frac{1}{m} F'(t) is solved by 2...
  31. V

    To Study Trajectory Of Shuttlecock In Badminton

    Okay, so I am a team member of a college robotics team, and this year we have quite challenging theme of playing badminton with robots, so for this reason we decided to track the position of shuttlecock in air. Now our approach was to predict the final position where it will land on ground from...
  32. samjohnny

    Solving for v: Seeking Guidance for Differential Equation

    Homework Statement Kindly see the attachment. Homework Equations The Attempt at a Solution As with all such questions, its in setting everything up that I'm having some trouble. I know that F = mdv/dt + vdm/dt. And also that F = R - m(t)g, but R = M0g. From here though I don't know how to...
  33. M

    Identity for Matrix*Vector differentiation w.r.t a vector

    I have J - matrix x and y - vector d [ J(x) y(x)] / dx I can multiply the matrix and vector together and then differentiate but I think for my application it would be better to find an identity like {d [ J(x) y(x)] / dx } = J(x) d y(x) / dx + d J (x) / dx y(x) I am not sure if this identity...
  34. B

    Can you cancel a function out of a differential equation?

    I saw this in http://en.wikipedia.org/wiki/Momentum_operator From equation 4 to 5, it seems that a function is canceled out from the partial derivatives, is this possible?
  35. M

    Solving a differential equation

    Homework Statement Solve (xy+y2+x2) dx -( x2 )dy = 0 Homework Equations to verify if exact http://upload.wikimedia.org/math/5/2/c/52cc749bb1c32abf1dccf613bd847a6e.pngM/[PLAIN]http://upload.wikimedia.org/math/5/2/c/52cc749bb1c32abf1dccf613bd847a6e.pngy =...
  36. KD1729

    MHB Identifying the Degree of a Differential Equation

    How Can we define the degree of differential equation ? What is the degree of \left(\frac{{d}^{2}y}{d{x}^{2}}\right)^{\!{2}}\left(\frac{dy}{dx}\right)^{\!{3}} +\left(\frac{dy}{dx}\right)^{\!{1}} =0 ??(Wondering)
  37. O

    Limits of Differential Equations

    Homework Statement I need help finding the limit of the differential equation. (dx/dt) = k(a-x)(b-x) that satisfies x(0)=0 assuming a) 0<a<b and find the limit as t->infinity of X(t) b) 0<a=b and find the limit as t->infinity of X(t) Homework Equations none The Attempt at a Solution I...
  38. S

    MHB Find the general solution of the given differential equation

    Find the general solution of the given differential equation.. (1+t^2)y' + 4ty = (1+t^2)^{-2} I'm kind of confused here on what to do... Do I want to do something like e ^{\int4t} dt and then multiply that through on both sides or do I need to do something different here..I'm not really sure...
  39. B

    L-R-C Series Circuits - Help With Differential Equation

    Hello, this is a maths problem that is related to a physics problem, but I think it's best posted here due to what I'm asking about. 1. Homework Statement \frac{d^{2}q}{dt^{2}} + \frac{R}{L} \frac{dq}{dt} + \frac{1}{LC}q = 0 is a differential equation describing how charge and current change...
  40. W

    Laplace transform for differential equation

    Homework Statement use laplace transforms to solve the differential equation y"+2y'+17y = 1 Homework Equations Initial conditions are y(0) = 0 y'(0) = 0 The Attempt at a Solution so it converts to Y(s) (s^2+2s+17) = 1/s which then ends up as; Y(s) = 1/s*1/(s^2+2s+17) i know i need to invert...
  41. electronic engineer

    Third order differential equation

    Hi all, I need to understand these differential equations specially moving from the second order to the third order because i couldn't understand how they got to the result, what was exactly the principle: $$ y'=f(x,y) $$ $$ y''=\frac{df}{dx}(x,y(x)) = f_{x}(x,y) + f_{y}(x,y)y' = f_{x}(x,y) +...
  42. H

    Modeling: Light irradiance and spatio-temporal decay

    Hello everyone! 1. Homework Statement I want to model the irradiance on a fluid element while it is flowing, from a fixed(above or below) light source that has an irradiance that shows first order decay with time and from x-naught to x-infinity . here are the assumptions: -Laminar flow at...
  43. L

    Differential equation for motion within stars

    I know: f = ma f = - GMm/r^2 a = -GM/r^2 can be easily derived, But, we've been given the differential equation of motion of gas within a star as: a = -GM/r^2 - 1/p * dP(r)/dr I was wondering where the - 1/p * dP(r)/dr term is derived from? I can't find it in my textbooks. Cheers
  44. Z

    Solving a differential equation

    I want to solve Equation (1). w is a constant: \begin{eqnarray} \text{Equation (1): }\frac{dx}{dt}=1+x^2w^2\end{eqnarray} and I have been told that it is solved by (2): \begin{eqnarray} \text{Equation (2): }x(t)=\frac{Ax(0)+B}{Cx(0)+D}\end{eqnarray} Problem I believe them, but before I keep...
  45. N

    Differential equation solution basic

    Ok so I tried to solve the following differential equation for y by every method (variable separable method/making to variable separable method technique/putting y/x=other variable/making homogeneous to solve equation to get y/x ).But I think I can't solve this particular equation by these...
  46. E

    Solving differential equation using Laplace Transform

    Homework Statement solve the following differential equation using Laplace transforms: y'' + 4y' + 4y = t^2 e^{-2t}, y_0 = 0, y'_0 = 0 y_0 and y'_0 are initial conditions. Homework Equations Using L to represent the Laplace transform, we have that L(y) = Y L(y') = pY - y_0 L(y'') =...
  47. MrOmar

    Need help solving a differential equation numerically

    Hello all, I am currently trying to solve a differential equation numerically. The equation is as follows: dv/dt = (u*q)/(m0-q*t) - g - (cd*ρ*A*0.5)*v2 If you haven't already guessed, it's the rocket equation with added gravity and drag. Now, I'm not even sure if that's what it's supposed to...
  48. S

    Did I set up my differential equation correctly?

    Homework Statement Shown in attachment The problem has been modified. All inputs and outputs are 5 gal/min. Pure water enters tank 1. Homework Equations System of equations The Attempt at a Solution Included on attachment.
  49. L

    How can you find this differential equation?

    This is actually a chemistry problem, but I feel it's more appropriate posted here, as I'm not having trouble with the chemistry but rather the mathematics. To avoid mixing subjects, I'll keep chemistry jargon out. We're given the formula for a component, and the problem request we give...
  50. Dethrone

    MHB How to Solve a Tough Differential Equation on a Calc I Exam?

    Solve: $$\sin\left({x}\right)y''+(y'^2-\sin^2\left({x}\right))^{1/2}y'^2-\cos\left({x}\right)y'=0$$ Initial conditions: $y(0)=0$ $y'(0)=1$ Keep in mind that this question was on a Calc I exam worth 5 marks, so please nothing crazy like reduction of order or anything...:D
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