Differential equation Definition and 1000 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.

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  1. Summer95

    I Confused about Finding the Green Function

    Suppose we have a differential equation with initial conditions ##y_{0}=y^{\prime}_{0}=0## and we need to solve it using a Green Function. Then we set up our differential equation with the right side "forcing function" as ##\delta(t^{\prime}-t)## (or with ##t^{\prime}## and ##t## switched I'm a...
  2. Reno Nza

    Differential equation for a vibration problem

    Homework Statement The data given was the acceleration of the component over time, as below: time | Acceleration (m/s²) 0 | 0 0,2 |3,61 0,4 |4,5 0,6 |5,4 0,8 |7,508 1 |12 - The Rigidity constat is 12600000 [N/m] (for the spring) The damping constant is 6500 [N/m] (for the damp)...
  3. chwala

    Differential Equation: Solving for x in terms of θ | (3+cos2θ)dx/dθ = xsin2θ

    Homework Statement Solve and obtain an expression for x in terms of θ given ## (3+ cos 2θ) dx/dθ = x sin 2θ## Homework EquationsThe Attempt at a Solution ...##dx/x = ((sin 2θ)/(3+ cos 2θ))dθ## , let ##u = (3+cos 2θ)⇒ du/dθ= -2 sin 2θ## thus ##∫dx/x = -1/2∫du/u ##...[/B]are my steps...
  4. P

    First order Partial differential equation

    Homework Statement Find a solution of $$\frac{1}{x^2}\frac{\partial u(x,y)}{\partial x}+\frac{1}{y^3}\frac{\partial u(x,y)}{\partial y}=0$$ Which satisfies the condition ##\frac{\partial u(x,y)}{\partial x}\big |_{y=0}=x^3## for all ##x##. The Attempt at a Solution I get the following...
  5. G

    Second order derivatives and resonance

    Homework Statement 2. Consider an electric circuit consisting of an inductor with inductance L Henrys, a resistor with resistance R Ohms and a capacitor with capacitance C Farads, connected in series with a voltage source of V Volts. The charge q(t) Coulombs on the capacitor at time t ≥ 0...
  6. Avi Nandi

    A Stochastic Differential equation: Dichotomous Process

    I am studying a dichotomous markov process. The master equation is given in this link https://en.wikipedia.org/wiki/Telegraph_process. I want to calculate the mean and correlation function given also in the link. But actually I can't make any progress. How from this master equation governing the...
  7. Jose Z

    Help Solving the following Differential equation

    Homework Statement Solve (x-1)y''-xy'+y=0 , given x>1 and y1=ex Homework EquationsThe Attempt at a Solution I've tried solving it by multiplying everythin for (1/x-1), so my equation looks like: y''-(x/x-1)y'+y=0 (because x-1 is unequal to 0) (1) so now the equation has the...
  8. Z

    Missing Power Solving the PDE (Solution included)

    Homework Statement Homework EquationsThe Attempt at a Solution integral[du]= Integral[xt ds] xt=18s2+3sT so, u=Integral[18s2+3sT] u=6s3+(3/2)s2T+C C=eT2 This is what I did and the solution is below. I'm unsure where the missing power on the (3/2)sT went in the u(s,t) equation.[/B]
  9. faradayscat

    Differential equation with power series

    Homework Statement Solve y''+(cosx)y=0 with power series (centered at 0) Homework Equations y(x) = Σ anxn The Attempt at a Solution I would just like for someone to check my work: I first computed (cosx)y like this: (cosx)y = (1-x2/2!+x4/4!+ ...)*(a0+a1x+a2x2 +...)...
  10. D

    Engineering Diode - RC circuit differential equation

    Homework Statement I'm having difficulty deriving the differential equation, this is what I have so far. In order to solve it, I will be using Matlab, and I'll be using the equation dy/dx ≈ (y(x + dx) - y(x))/dx. Is my derivation correct so far? Homework Equations In picture. The Attempt at a...
  11. Dusty912

    How Can Phasors Represent the Function g(t) in Complex Form?

    Homework Statement Given a function g(t)=acosωt + bsinωt, where a and b are constants, show that g(t) is the real part of the complex function: keiΦeiωt for some k and Φ Remark: the complex expression keiΦ is called a phasor. If we know that g(t) has the form kcos(ωt+Φ) then we need know only...
  12. J

    Two blocks collide with spring in-between.

    Homework Statement Block 1 (mass 2 kg) is moving rightward at 10 m/s and block 2 (mass 5kg kg) is moving rightward at 3 m/s. The surface is frictionless, and a spring with spring constant of 1120 N/m is fixed on the left side on block 2. When the blocks collide, the compression of the spring is...
  13. NihalRi

    Simple differential equation problem

    Homework Statement Find constants a and b such that y=ax + b is a solution to the differential equation dy/dx = 4x - 2y Homework EquationsThe Attempt at a Solution I already have the solution that is: a=4x-2 (ax+b) (I'm fine with this part) a = (4-2a) x - 2b (what happened here?) 4-2a=0...
  14. Donghyun Kim

    A Is it possible to solve this differential equation?

    Hello. I'm solving the second order nonlinear ODE, and I'm not sure that it's possible or not. Please help. A_0, B_0, C_0, a, b, c, e, p is all known constant. dA/dt = -(a+b*C)*B dB/dt = -p*c*A dC/dt = -(1-p)*e*A then I want to get a solution like A(t) = function of (A_0, B_0, C_0, a, b, c...
  15. E

    Linear Differential equation problem

    Homework Statement Solution of the differential equation ##(\cos x )dy = y (\sin x - y) dx , 0 < x < \dfrac{\pi}{2} ## is Homework EquationsThe Attempt at a Solution Only separation of variables, homogenous and linear DEs are in the syllabus, therefore it must be one of those. It obviously...
  16. K

    Schrodinger Equation/verify solution

    Homework Statement Trying to use change of variables to simplify the schrodinger equation. I'm clearly going wrong somewhere, but can't see where. Homework Equations [/B] Radial Schrodinger: -((hbar)2)/2M * [(1/r)(rψ)'' - l(l+1)/(r^2) ψ] - α(hbar)c/r ψ = Eψ The Attempt at a SolutionWe're...
  17. K

    IalChange of variables/verifying solution

    Homework Statement Trying to use change of variables to simplify the schrodinger equation. I'm clearly going wrong somewhere, but can't see where. Homework Equations [/B] Radial Schrodinger: -((hbar)2)/2M * [(1/r)(rψ)'' - l(l+1)/(r^2) ψ] - α(hbar)c/r ψ = Eψ The Attempt at a SolutionWe're...
  18. A

    I Why does the amplitude of an undamped driven oscillator not vary with time?

    The equations I'm getting when I solve the differential equations seem to imply that the amplitude of oscillation does not vary in time. For example, if I have x'' + ω02x = cos(ωt) If we suppose that ω≠ω0, then the general solution should look something like: x(t) = c1cos(ω0t) + c2sin(ω0t)...
  19. A

    I Does the Amplitude of an Undamped Driven Oscillator Stay Constant Over Time?

    The equations I'm getting when I solve the differential equations seem to imply that the amplitude of oscillation does not vary in time. For example, if I have x'' + ω02x = cos(ωt) If we suppose that ω≠ω0, then the general solution should look something like: x(t) = c1cos(ω0t) + c2sin(ω0t)...
  20. H

    B Recast a differential equation with a change of variable

    I obtained an extra factor of ##\frac{1}{l^2}## in the first term on the LHS of (3.34). From (3.33), LHS ##=u^2\frac{d}{d\theta}(\frac{u^2}{m}\frac{du}{d\theta}\frac{dr}{du})-\frac{l^2u^3}{m}## ##=u^2\frac{d}{d\theta}(-\frac{1}{m}\frac{du}{d\theta})-\frac{l^2u^3}{m}##...
  21. K

    Oscillations of a free hanging chain

    Homework Statement I am trying to find an equation for a free hanging chain of mass m and length L. The chain is hanging vertically downwards where x is measured vertically upwards from the free end of the chain and y is measured horizontally. Homework Equations [/B] I derived this...
  22. F

    Form differential equation by substituition

    Homework Statement dy/dx = (x-y+2) / (x+y-2) , by using x=X and y=Y+2 , form a differential equation . Homework EquationsThe Attempt at a Solution the author gave the answer as (Y^2) +(2XY) -(X^2)+A= 0 , while my answer is (Y^2) +(2XY) -(X^2)+A(X^4)= 0 is there anything wrong with my answer?
  23. Mrinmoy Naskar

    A What is the significance of the constant m in the Hermite differential equation?

    y" - 2xy' + my = 0 this is well known hermite diff eqn. now can anyone tell me what kind of conts is m?? what is the suitable value of m??
  24. A

    I Conceptual Question: Vector-Matrix Differential Equation

    Hi I'm just having trouble wrapping my head around differential equations with matrices and vectors... For example: let y be a vector. let A(t) be an nxn matrix. I have the differential equation: dy/dt = A(t)y So I think I understand why the solution is y = ceA(t) But I'm having trouble...
  25. S

    A Nonlinear first order Differential equation

    I need to solve the well known momentum equation in 3D cylindrical coordinates: ρ(∂v/∂t +(v.∇)v)=A where A and the velocity v are both local vector variables. I am actually looking for the stationary solution to the equation, i.e. no ∂/∂t term) I have tried evolving the velocity and tried...
  26. chwala

    Differential equation tree height

    1. Homework Statement A tree is planted as a seedling of negligible height. The rate of increase of its height , in metres per year is given by ##0.2√(25-h)## a. explain why tree can't exceed 25 metres. answer⇒ ##dh/dt=0## when h=25 b. express t as a function of h answer⇒...
  27. A

    Solving Differential Equation numerically

    Homework Statement I am supposed to write a script that can solve the Schrödinger equation on a nonuniform grid. Homework Equations Finite element approximation to the second derivative as in: https://www.physicsforums.com/threads/nonuniform-finite-element-method.857334/#post-5382329 The...
  28. G

    General solution to differential equation

    Homework Statement Find the general solution to differential equation y''-6y'+9y=e^x((2x+1)\cos x+(x+3)\sin x) Homework Equations -Non homogeneous differential equation -Homogeneous differential equation with constant coefficients -Method of undetermined coefficients The Attempt at a Solution...
  29. G

    Dy/dx = (y^2) /x , form differential equation

    Homework Statement my ans is lnx = (-1/y) + c (-1/y) = lnx -c y = -1/ (lnx -c ) , but the answer given is (-1/ln x )+ C , how to get the answer given ? Homework EquationsThe Attempt at a Solution
  30. R

    Dirac Delta Function in Differential Equation

    Homework Statement Find the solution to: $$\frac{d^2}{dt^2} x + \omega^2 x = \delta (t)$$ Given the initial condition that ##x=0## for ##t<0##. First find the general solution to ##t>0## and ##t<0##. Homework Equations The Attempt at a Solution This looks like a non-homogeneous second...
  31. A

    Is the Hypothesized Law of Gravity for a Falling Body's Speed Accurate?

    Homework Statement The speed of a falling body might be based on the observation that the velocity of a falling object seems to increase the further it has fallen. Model the hypothesis "The speed of a falling object is proportional to the distance it has fallen" as a differential equation...
  32. M

    Need help solving a differential equation

    Are there any known analytical method to solve the equation $$ A\frac{d^2f(x)}{dx^2}+B\frac{df(x)}{dx}+Ce^{igx}f(x) = 0\hspace{1cm}? $$ All quantities appearing in that equation are complex except for ##g## and ##x##.
  33. D

    I How to Approach Solving 2y+(y')^2+ax^b=0?

    I tried to solve this differential equation: 2y+(y')^2+ax^b=0 ...but don't know what to do with it. Don't know what variable substitution to use. Tried Taylor series, but I get horrible nonlinear equations for the coefficients. Tried Mathematica, but it doesn't answer anything: In[20]:=...
  34. kostoglotov

    Engineering Does this (textbook) circuit diagram contain a mistake?

    Homework Statement [/B] This circuit solves some differential equation, the question is asking for the equation based on the circuit diagram imgur link: http://i.imgur.com/b1npQSK.pngHomework EquationsThe Attempt at a Solution I refer to the input terminal for -f(t)...the signal from it...
  35. EastWindBreaks

    Integrating Factor: Homework Help

    Homework Statement hello, I was reading through the textbook and I have a hard time to understand this part: Homework EquationsThe Attempt at a Solution haven't been dealing with derivatives for a while, i don't understand how it got ln |u(t)| from the first equation. Am I treating the...
  36. Shackleford

    Derive differential equation that describes temperature

    I'm taking an engineering heat transfer course as an elective. 1. Homework Statement Copper tubing is joined to a solar collector plate of thickness t, and the working fluid maintains the temperature of the plate above the tubes at To. There is a uniform net radiation heat flux q”rad to the...
  37. H

    Differential equation of dy/dx = (2x-y+4)/ (4x-2y +1)

    Homework Statement I'm asked to use the transformation of v= 2x-y to solve dy/dx = (2x-y+4)/ (4x-2y +1) the answer given is (2/9)(6x-3y-2) +(7/9)ln(6x-3y-2) = x +c , i got (2/9)(6x-3y) +(7/9)ln(6x-3y-2) = x +c , what's wrong with my working ? Homework EquationsThe Attempt at a Solution
  38. F

    Solve the following differential equation

    Homework Statement Solve the differential equation. -x2(dy/dx) + xy = x2y2 * sin(x) Homework Equations None. The Attempt at a Solution I first figured out that this was a Bernoulli's equation. I distributed the x2 to make it simpler. From there I divided everything by y2 to get y-2(dy/dx) +...
  39. H

    First order differential equation

    Homework Statement dy/dx = (x +y) / (x-y) , i am asked to find the first order differential equation , but the ans i gt is different from the ans given Homework EquationsThe Attempt at a Solution
  40. H

    How to Solve a Non-Separable Differential Equation?

    Homework Statement i was given dy/dx = -(4x+4y) / (4x+4y-2) , v= x +y . I have tried to do ( as in the photo) , but I didn't get the ans , in the last line of my working , the V and x are not separable , which part of my working is wrong ? Homework EquationsThe Attempt at a Solution
  41. T

    Differential equation help

    I understand what is in the picture http://postimg.org/image/u5ib33kzb/ but the book goes on to say that the solution is thus of the form ## X_n = a_n sin \frac{n \pi x}{l} ## How does putting ##β=σ^2=\frac{n^2π^2}{l^2}## into (6.37) result in that?
  42. C

    Why is differential equation equal to sum of partials?

    My current understanding of differential equations is extremely shaky, and my vocabulary is probably very incorrect, but I'm curious about something I've recently seen in some Khan Academy videos (specifically this one) and in other situations with differential equations. It seems that the...
  43. kostoglotov

    Jordan Chains to solve x'=Ax, complex-valued.

    This isn't a homework help question, I am asking about a general case as the text hasn't fleshed out it's explanation enough for me. So, I've just seen how Jordan Chains can be applied to solve some applied problems involving solving x'=Ax. My question is about what they specifically mean...
  44. E

    Project idea in Numeric Analysis

    hey, need an idea for a paper on differential equations, which can be implemented in two or three days work. also it will be nice if that paper has graphs on it. any help would be most welcome thx
  45. Dusty912

    Check My Work for solving this Differential Equation? :)

    Homework Statement Question: Solve this differential equation dy/dt=1+y2 and y(π/4)=-1 Homework Equations The Attempt at a Solution dy/dt=1+y2 and y(π/4)=-1 dy/dt=1+y2 dy=(1+y2)dt dy/(1+y2)=dt ∫dy(1+y2)=∫dt tan-1(y)=t+c y(t)=tan(t)+c y(π/4)=tan(π/4)+c -1=1+c -2=c answer: y(t)=tan(t)-2
  46. ATY

    Elliptic partial differential equation

    Hey guys, so my professor told me to take a look at an equation, because he thinks that there is a mistake. We are basically talking about exercise 6.3 (on last image). The pictures will show you the text, so that you have all the information, that I have http://puu.sh/mrNDl/ec19cdff63.png...
  47. G

    Deriving hermite differential equation from schrødinger harm oscillator

    Homework Statement I am trying to obtain the hermite polynomial from the schrødinger equation for a har monic oscillator. My attempt is shown below. Thank you! The derivation is based on this site: http://www.physicspages.com/2011/02/08/harmonic-oscillator-series-solution/ The Attempt at a...
  48. G

    Why 2nd order differential equation has two solutions

    Hello. I forgot the reason why 2nd order differential equation has two independent solutions. (Here, source term is zero) Why 3 or 4 independent solutions are not possible? Please give me clear answer.
  49. Alexiy

    Find the differential equation and velocity

    Homework Statement Homework Equations 3. The Attempt at a Solution [/B] Hello guys,I posted images since its easier to write equations.Please can someone help me check this, if this is correct so far, then i should be able to find the velocity at C, using kinetic energy?
  50. L

    Differential equation mixing problem

    Homework Statement A vat with 2000L of beer contains 4% alcohol (by volume). Beer with 6% alcohol is pumped into the vat at a rate of 20L/min and the mixture is pumped out at the same rate. What is the percentage of alcohol after an hour? Homework EquationsThe Attempt at a Solution At ##t =...
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