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. T

    Differential Equation (2nd Order Linear)

    Homework Statement (x2)d2y/dx2 + 2*x*(dy/dx) + w2*x2*y=0 Where w is a constant Homework Equations The Attempt at a Solution I am having a really hard time figuring out how to solve this. Usually for second order linear ODEs I start with assuming a solution of form y=eλx...
  2. D

    Differential equation (substitution)

    Homework Statement for this question , i 've got my positive 9 but i got -64 , can anyone tell me which part is wrong? Question : https://www.flickr.com/photos/123101...3/13907725466/ Wroking : https://www.flickr.com/photos/123101...n/photostream/ Homework Equations The...
  3. K

    Solving Differential Equation: ln |9/64| = k

    i got k = ln |9/64| then how can the next step using ln 0 doesn't make sense. what should i do?
  4. M

    Simplifying a differential equation

    When I multiply out the first line I end up with an extra (dA/dx)*(dσ/dx). Can someone please show me how i get from the first line to the second. Thanks
  5. H

    Differential Equation Initial Value Problem

    Homework Statement I managed to work this problem all the way through, but I am in no way certain of my answer. I'd greatly appreciate any insight! Find the solution of the initial value problem. y'''+4y'=x, y(0)=y'(0)=0, y''(0)=1 Homework Equations Just for clarification...
  6. R

    Solving the differential equation of an object oscillating in water.

    I have a differential equation to solve below on the motion of an object oscillating in water with a restoring force equal to -Aρgx and a damping force equal to -kv^2. ma+kv^2+Aρgx=0 K, A, ρ and g are constants and I need to solve the equation for x. a (acceleration). v (velocity). x...
  7. W

    Numerical Solution of Differential equation

    The nonlinear oscillator y'' + f(y)=0 is equivalent to the Simple harmonic motion: y'= -z , z'= f(y) the modified Symplectic Euler equation are y'=-z+\frac {1}{2} hf(y) y'=f(y)+\frac {1}{2} hf_y z and deduce that the coresponding approximate solution lie on the family of curves...
  8. A

    Simple Differential Equation Modeling

    Homework Statement I am trying to model using differential equations rain falling from the sky, hitting the pavement, and then running off the pavement into the ground. I'm a little rusty on differential equations and I'm just wondering if my answer is correct. In this model I'm assuming that...
  9. R

    Advanced numerical solution of differential equation

    Show that the explicit Runge-Kutta scheme \begin{equation} \frac {y_{n+1} -y_{n}}{h}= \frac{1}{2} [f(t,y_{n} + f(t+h, y_{n}+hk_{1})] \end{equation} where $k_{1} = f(t,y_{n})$ applied to the equation $y'= y(1-y)$ has two spurious fixed points if $h>2$. Briefy describe how you would...
  10. M

    MHB Parabolic 2. order partial differential equation

    Hey! :o I have to find the Normal form of the 2.order partial differential equation. I am not sure if my solution is correct.. The differential equation is: $ u_{xx}-4u_{xy}+4_{yy}-6u_x+12u_y-9u=0$ $a=1, b=-2, c=4$ $b^2-ac=4-4=0 \Rightarrow $ parabolic $\frac{dy}{dx}=\frac{1}{a}(b \pm...
  11. A

    Solving Partial Differential Equations: Strategies and Techniques

    I have, as is normal for anyone working in physics, come across a differential equation describing the system I am looking it. Now I know little about solving partial differential equations, and indeed I am not even sure if an analytical solutions exists for my equation, but here it is anyways...
  12. M

    MHB Normal form of the differential equation

    Hey! :o I have the following exercise: Write in the normal form the differential equation $$u_{xx}+\frac{2y}{x}u_{xy}+\frac{y^2}{x^2}[(1+y^2)u_{yy}+2yu_y]=0$$ Hint: You can suppose that the one new variable is given by $\xi=x$ I have done the following: $a=1, b=\frac{y}{x}...
  13. S

    How do I solve differential equation?

    Homework Statement Solve ##y''+y-sinx=0##. Homework Equations The Attempt at a Solution I am actually working on variational problems which brought me to this differential equation. I thought that taking ##y=Asinx+Bcosx## would solve it, yet it does nothing useful. In other...
  14. D

    Differential equation right hand function

    Homework Statement The question specifies the auxiliary equation given is (D^2 + D - 2) = (e^x)/(x) the method of variation of parameter must be used to find the particular solution to the right hand function. then finally the general soultion should be stated.Homework Equations variation of...
  15. S

    Differential equation with Fourier Transform

    Homework Statement Without solving the differential equation, find the differential equation that solves Fourier transformation of given differential equation for ##a>0##. a) ##y^{'}+axy=0## b) For what ##a## is the solution of part a) an eigenfunction of Fourier Transform Homework Equations...
  16. Ravi Mohan

    Solving second order coupled differential equation

    How do we solve a system of coupled differential equations written below? -\frac{d^2}{dr^2}\left( \begin{array}{c} \phi_{l,bg}(r) \\ \phi_{l,c}(r) \\ \end{array} \right)+ \left( \begin{array}{cc} f(r) & \alpha_1 \\ \alpha_2 & g(r)\\ \end{array} \right).\left( \begin{array}{c}...
  17. G

    The spectrum of a bounded differential equation

    is it possible to work backwards from a spectrum to which operator?
  18. B

    Power Series Solution of Differential Equation

    Homework Statement (x^2)y' = y Homework Equations The Attempt at a Solution Plugging in series everywhere I get the equation \sum na_{n}x^{n+1} = \sum a_{n}x^{n}. I try to set the coefficients for the corresponding powers equal, but when I do I don't get the correct answer. I also...
  19. E

    Second order differential equation

    Homework Statement http://books.google.co.uk/books?id=93b3cjVJ2l4C&lpg=PA135&ots=8OtqgKwrQ2&dq=%22Two%20particles%20are%20connected%20by%20a%20spring%20of%20spring%20constant%20k%22%20and%20zero%20equilibrium%20%20length&pg=PA136#v=onepage&q&f=false Homework Equations All in the link...
  20. D

    How to start Differential equation

    Mod note: Thread moved from technical math section. The OP has already been notified that this is not a suitable start to a request for homework help. can anyone teach me how to start it? i really have no idea.. PART A
  21. P

    Second Order Linear Differential Equation

    Hey, I'm not sure how to even approach this problem. It's not a simple ODE. Basically, I want to find the solution for Θ in terms of ε. The equation is \frac{1}{ε}*\frac{d}{dε}*(ε*\frac{dΘ}{dε})-β^{2}Θ=0 I tried to move the B^2 to the other side and I wasn't able to solve it that way. I...
  22. L

    Is y = sqrt(x-1) a Solution to 2yy' = 1?

    The question ask to determine whether y = φ(x) = sqrt ( x - 1 ) is a solution of the differential equation 2yy' = 1.
  23. J

    Forced oscillations: solution of Differential Equation at resonance

    Homework Statement Following a worked example in my book, I have been trying to get a solution for the equation \frac{d^2u}{dt^2} + \frac{k}{m}u = Fcos\omega t The book says that at resonance, i.e. when \omega_0 (the natural frequency) = \omega (the forcing frequency), the term F cos\omega...
  24. Saitama

    MHB Solving a first-order differential equation

    Problem: Solve the differential equation: $$\left(\frac{1}{x}-\frac{y^2}{(x-y)^2}\right)\,dx+\left(\frac{x^2}{(x-y)^2}-\frac{1}{y}\right)\,dy=0$$ Attempt: Let $$M=\left(\frac{1}{x}-\frac{y^2}{(x-y)^2}\right)$$ and $$N=\left(\frac{x^2}{(x-y)^2}-\frac{1}{y}\right)$$ I noticed that...
  25. R

    MHB Advanced topic in numerical solution of differential equation

    Investigate the stability of the PECE method where P=Predictor : y_(n+1) = y_n + hf(y_n) C=Corrector: y_(n+1) = y_n + h [(1-θ) f(y_n) + θ f(y_(n+1))], (0<θ<1) and E is the evaluation step. => substituting the predictor into corrector gives:y_(n+1) = y_n + h [(1-θ) f(y_n) + θf( y_n+ h f y_n...
  26. D

    Engineering Solving RL Circuit Differential Equation: Initial Conditions

    Homework Statement Consider the RL circuit shown in the figure. Assume that the current ##i(t)## has reached a steady state with the switch at position ##A##. At time ##t = 0##, the switch is moved from position ##A## to position ##B##. http://imgur.com/dRIOrp0 If I use the image button...
  27. D

    MHB Solving the RLC Circuit Differential Equation

    Determine the differential equation relating \(v_i(t)\) and \(v_0(t)\) for the RLC circuit in the figure. Would this just be \[ v_i(t) = 3i + \frac{di}{dt} + 2\int i(t)dt \] but \(v_0 = 2\int i(t)dt\). Do I need write it as \(v_0\) or as \(2\int i(t)dt\)?
  28. J

    Autonomous differential equation

    Homework Statement A pond forms as water collects in a conical depression of radius a and depth h. Suppose water flows in at a constant rate, k and is lost through evaporation at a rate proportional to the surface area. I was wondering whether anyone could give me some guidance on this...
  29. J

    Solving First Order Differential Equation

    Homework Statement I have been trying to solve this equation but keep coming to the same solution, which according to my book is not the correct one. Is anyone able to point out what I am doing wrong? \frac{dy}{dt}-\frac{1}{2}y=2cos(t) The Attempt at a Solution To solve, use...
  30. P

    Markov model and its differential equation

    Hi guys, I have problem in constructing the corresponding differential equation for a markov model: Given a markov model as shown S1→(a1)→S2→(a2)→S3 S1←(b1)←S2←(b2)←S3 where S1 to S3 are three different states of the system and ai and bi are the forwards and backwards rate constant for...
  31. S

    How do I get d^2 y/dx^2 for a Cauchy-Euler, differential equation?

    Homework Statement How do I get d^2 y/dx^2 for a Cauchy-Euler, differential equation? Basically, how do I derive d^2 y/dx^2, as given in the following link (since I don't want to just memorize that equation)?: http://www.sosmath.com/diffeq/second/euler/euler.html Homework Equations *...
  32. S

    1st order differential equation

    Find the general solution of the first order differential equation (y+x^{2}y)\frac{dy}{dx}=3x+xy^{2}, with y(1)=1. My attempt: \frac{y}{3+y^{2}}dy=\frac{x}{1+x^{2}}dx ∴ \frac{1}{2}\int \frac{2y}{3+y^{2}}dy=\frac{1}{2}\int \frac{2x}{1+x^2}dx...
  33. B

    Differential Equation: Solving for h(t) with Constant a, b, and c

    Homework Statement a*dh(t)/dt + h(t) = b * sin(c*t) How can I get the equation for h(t) from this equation?? a,b,c are constant
  34. tom.stoer

    Inexact differential equation with unknown function M(t)

    I am looking for a general expression for an integrating factor μ(x,t) to solve the following diffential equation for x(t) \frac{dx}{dt} = \frac{x - f}{x} f = f(t) is an arbitrary function of t with f > 0 and df/dt < 0 Any ideas?
  35. P

    Solving a differential equation

    Homework Statement Find the general solution: 2xy \frac{dy}{dx} = y + x^4Homework Equations The Attempt at a Solution I have tried to solve this as a linear first order equation, a Bernoulli equation, and an exact equation. I'm not sure how to approach this, any ideas?
  36. E

    General Solution / Differential Equation

    Homework Statement Find the general solution x(t) to the following differential equation: dx/dt = 2t/5xHomework Equations dx/dt = 2t/5x The Attempt at a Solution My solution is: ∫5xdx = ∫2tdt (5/2)x^2 = t^2 + C x^2 = (2/5)(t^2 + C) x = +-√[(2/5)(t^2 + C)] However, when I put the problem in...
  37. paulmdrdo

    MHB Differential equation population growth problem

    A bacterial population B is known to have a rate of growth proportional to B itself. If between noon and 2pm the population triples, at what time no controls being exerted, should B becomes 100 times? what it was at noon? using this formula $\displaystyle P(t) \;=\;P_oe^{kt}$ please help me...
  38. paulmdrdo1

    MHB Differential Equation of Unknown Type

    can you help me solve this, $\displaystyle (x^2+y^3+1)dx+x^4y^2dy=0$ I can't see any particular exact D.E form here. please help.
  39. C

    Inhomogenous differential equation of second order

    Homework Statement y^{\prime\prime}+y=\frac{1}{\sin x} Homework Equations The Attempt at a Solution I solved the homogenous equation: y=C_1\sin x+C_2\cos x, and then I tried to use method of variable the constant. But the equation system is rather hard. Do you know any other...
  40. C

    Calculate differential equation

    Homework Statement y=xy^\prime-\left(y^\prime\right)^2 Homework Equations The Attempt at a Solution Unfortunately, I do not have any good idea. I tried y=xt(x), but the equation only became worse.
  41. paulmdrdo1

    MHB Help Me Solve This Differential Equation!

    please help me continue solving this, $\displaystyle \frac{dy}{dx}=\ln(x)-\ln(y)+\frac{x-y}{x+y}$ this is where I can get to, $\displaystyle \frac{dy}{dx}=\ln(\frac{x}{y})+\frac{x-y}{x+y}$ multiplying the 2nd term by $\frac{1}{x}$ $\displaystyle...
  42. U

    Find the general solution of the differential equation

    Homework Statement The equation: \frac{dx}{dt}=\frac{t^2+1}{x+2}. Where the initial value is: x(0) = -2. Homework Equations I believe you have to use the method of seperations of variables. The Attempt at a Solution So I multiplied both sides with x+2. Then I integrated...
  43. B

    MHB What Is the Differential Equation for Given Solutions?

    find the desired equation. a.) $\displaystyle y=c_1+c_2e^{3x}$ taking two derivatives $\displaystyle \frac{dy}{dx}=3c_2e^{3x}$ $\displaystyle \frac{d^2y}{dx^2}=9c_2e^{3x}$ b.) $\displaystyle y=c_1e^{ax}\cos(bx)+c_2e^{ax}\sin(bx)$ a and b are parameters. can you help me continue with the...
  44. paulmdrdo

    MHB Meaning of Symbols in Differential Equation

    just want to know what these symbols mean $\displaystyle M(x,y)\,dx+N(x,y)\,dy=0$ $\displaystyle \frac{dy}{dx}=f(x,y)$ $\displaystyle F(x,y,y'...y^n)=0$ what's M and N and the ordered pair (x,y) mean here. I don't understand my book. please explain.
  45. R

    Linear non-exact differential equation made exact

    Homework Statement Find the general solution: (y+1) dx + (4x - y) dy = 0 Homework Equations dy/dx + P(x)y = Q(x) (standard form) e^(∫ P(x) dx) (integrating factor) The Attempt at a Solution This exercise is in the chapter on linear equations, making non-exact equations exact. So I know I...
  46. L

    Differential equation formation

    Homework Statement If α is an arbitrary constant and a a fixed constant show that xcos α + ysin α = a is the complete primitive of the equation (y - xdy/dx)^2 = a^2( 1 + (dy/dx)^2) Homework Equations The Attempt at a Solution FIrst I found the first derivative by...
  47. maistral

    Partial differential equation discretization. HELP D:

    So figuratively, I'm trying to win a nuclear war with a stick. :smile: I did not take any course in PDEs, I just self-studied some of them, and now I'm toast. :smile: First, please feel free to hurl rocks at me if my simplification is incorrect...
  48. A

    Find the differential equation or system of differential equations

    Find the differential equation or system of differential equations *** Find the differential equation or system of differential equations assoicated with the following flows a) ##\phi_t (x) = \frac{x}{\sqrt{1-2x^2t}} ## on ##{\mathbb R} ## b) ##\phi_t (x,y) = (xe^t, \frac{y}{1-y^t}) ## on...
  49. C

    Numerical solution to a partial differential equation

    Homework Statement Solve the inhomogenous partial differential equation \frac{∂^{2}u}{∂t^{2}}-\frac{∂^{2}u}{∂x^{2}}=-6u^{5}+(8+4ε)u^3-(2+4ε)u by using the NDSolve function in Mathematica for the interval [0,10] x [-5,5]. Homework Equations Initial conditions: u(0,x)= tanh(x)...
  50. Saitama

    MHB Solving a differential equation

    Problem: Solve: $$\frac{x\,dx-y\,dy}{x\,dy-y\,dx}=\sqrt{\frac{1+x^2-y^2}{x^2-y^2}}$$ Attempt: I rewrite the given differential equation as: $$\frac{(1/2)d(x^2-y^2)}{x^2d(y/x)}=\sqrt{\frac{1+x^2-y^2}{x^2-y^2}}$$ I thought of using the substitution $x^2-y^2=t^2$ but that doesn't seem to help...
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