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. Boltzman Oscillation

    Help derive this differential equation?

    Hello I need to derive this equation from Grittfith's quantum book $$ \frac{d^2y}{dr^2} = r^2y$$ I know I can use the characteristic equation: $$m^2 = r^2 \rightarrow y = e^{r^2}$$ but the answer should be: $$y=Ae^{\frac{-r^2}{2}} + Be^{\frac{r^2}{2}}$$ I know from Euler's formula that...
  2. D

    MATLAB Partial Differential Equation - solve with Matlab

    Help please, I need to solve this differential equation x\frac{\partial^2 U}{\partial x^2}+y\frac{\partial^2 U}{\partial y^2}=aU in Matlab (where "a" is a constant parameter, it can be taken by any), I wanted to use the Partial Differential Equation Toolbox, but I ran into a problem, the...
  3. GlassBones

    Change of variables on autonomous systems solutions

    Homework Statement Given that ##x=\phi (t)##, ##y=\psi(t)## is a solution to the autonomous system ##\frac{dx}{dt}=F(x,y)##, ##\frac{dy}{dt}=G(x,y)## for ##\alpha < t < \beta##, show that ##x=\Phi(t)=\phi(t-s)##, ##y=\Psi(t)=\psi(t-s)## is a solution for ##\alpha+s<t<\beta+s## for any real...
  4. echomochi

    Finding an implicit solution to this differential equation

    Homework Statement Find an equation that defines IMPLICITLY the parameterized family of solutions y(x) of the differential equation: 5xy dy/dx = x2 + y2 Homework Equations y=ux dy/dx = u+xdu/dx C as a constant of integration The Attempt at a Solution I saw a similar D.E. solved using the y=ux...
  5. R

    How do you solve a differential equation with complex numbers?

    Homework Statement Homework Equations euler ##e^{ix} = cos(x) + i*sin(x)## ##e^{-ix} = cos(x) - i*sin(x)## The Attempt at a Solution I'm starting with differential equations and I'm trying to understand this solution including complex numbers: First we determine the zeros. I understand that...
  6. BesselEquation

    Homogeneous Differential Equation

    Homework Statement Solve the following differential equation: y' = y / [ x + √(y^2 - xy)] 2. The attempt at a solution Using the standard method for solving homogeneous equations, setting u = y/x, I arrive at the following: ± dx/x = [1±√(u^2-u) ]/ [u√(u^2-u)] which in turn, I get the...
  7. Z

    Differential Equation with an Initial condition

    Homework Statement x(dy/dx) = 3y +x4cos(x), y(2pi)=0 Homework Equations N/A The Attempt at a Solution I've tried a couple different ways to make this separable, but you always carry over a 1/dx or 1/dy term and I can never fully separate this. I've also tried to do a Bernoulli differential...
  8. bob14

    Solving a Second-Order Nonlinear Differential Equation

    Homework Statement Hi, I'm trying to calculate the formula for the position vs. time of a rocket landing from an altitude of 100km. I'm neglecting a lot of forces for simplification but basically, I want to solve ##F_{net} = Drag - mg##. Homework Equations Drag Force: D = ## \frac {C_dAρv^2}...
  9. J

    B Why the Fourier series doesn't work to solve any differential equation?

    I know this may sound as a stupid question but I would like to clarify this. An arbitrary function f can be expressed in the Fourier base of sines and cosines. My question is, Can this method be used to solve any differential equation? You plug into the unkown function the infinite series and...
  10. N

    Partial Differential Equation with variable coefficients

    Homework Statement This question relates to a very large project I have been assigned in a course on mathematical methods in structural engineering. I have to solve the following equation, in a specific way: (17) Now we have to assume the following solution: (18) It wants me insert...
  11. SpaceIsCool

    Can This Differential Equation Be Solved by Separation of Variables?

    Homework Statement We solved the differential equation (2.29), , for the velocity of an object falling through air, by inspection---a most respectable way of solving differential equations. Nevertheless, one would sometimes like a more systematic method, and here is one. Rewrite the equation...
  12. J

    I with simple Differential Equation.

    Homework Statement Hello, I'm trying to go back to school and haven't done any math in awhile, as such, my skills are terribly out of practice. I am unable to arrive at the book's solution and suspect I am forgetting a simple algebraic trick and would like somebody to show me it and explain the...
  13. J

    MHB 2nd order differential equation - equation of motion

    Hi There is an example in my textbook worded as follows; A particle of mass 2kg moves along the positive x-axis under the action of a force directed towards the origin. At time t seconds, the displacement of P from O is x metres and P is moving away from O with a speed of v ms^-1. The force has...
  14. chwala

    Failure of uniqueness for this first-order differential equation

    Homework Statement how do we establish failure of uniqueness on this first order differential equation ## y(x)= x y'+(y')^2##Homework EquationsThe Attempt at a Solution [/B] general solutions are ## y= cx^2+c^2## where c = constant and ## y= -0.25x^2## ## -0.25x^2+cx+4c^2=0## ##x= -2c ⇒...
  15. Teri

    Differential equation - delta function

    Moved from technical math section, so is missing the homework template. How to solve this equation please? I found charakteristic roots ia ##\pm \sqrt{-a^{-k^2}}##. Thank you Moderator note: Edited the LaTeX above to show the exponent correctly.
  16. barryj

    A Hint as to how to solve an integral differential equation

    I know how to solve a differential equation using Eulers method but what if the equation has an integral part? i.e. a RLC electrical circuit. Vsource = iR + L di/dt + (1/C)int i dt can this be done? a link to how to solve this would be helpful.
  17. E

    I How do I find the integrating factor for a differential equation?

    So I've been Stuck On this Equation Trying to find the integrating factor (im not sure if it has one) appreciate the help
  18. WMDhamnekar

    MHB Difficult first order linear differential equation

    Hello, I want to solve the following differential equation. $y'=\dfrac{x^3-y^3}{x-y}$. How to solve it?
  19. W

    Finding the parametric equation of a curve

    Homework Statement Parameterize the part of the curve which allows an equilateral triangle, with the height 3R, to roll from one vertex to the next one, while its center travels at a constant height. Homework Equations I will include some pictures to show what I'm doing The Attempt at a...
  20. T

    I Solution to a second order differential equation

    I have currently been reading a book called 'Mathematical Methods In Physical Sciences'. Whilest I was looking at the differential section I came across a differential which I have never thought about before, which is of the form...
  21. H

    Partial Differential Equation Mathematical Modelling

    Salutations, I have been trying to approach a modelling case about organism propagation which reproducing with velocity $$\alpha$$ spreading randomly according these equations: $$\frac{du(x,t)}{dt}=k\frac{d^2u}{dx^2} +\alpha u(x,t)\\\ \\ u(x,0)=\delta(x)\\\ \lim\limits_{x \to \pm\infty}...
  22. Edge5

    I Solution of Quantum differential equation

    (I think I couldn't add the image) you can see my answer in link https://pasteboard.co/HPKZ6KD.jpg (Please first see my answer in the link) But in answer it is φ= Asin(kx) + Bcos(kx) I know that euler formula is eix = cosx +isinx But I can't get this answer can you help me?
  23. Phys pilot

    Critical points of a diff eq system

    Homework Statement I have to calculate the critical points of the following system. $$x'=cx+10x^2$$ $$y'=x-2y$$ The Attempt at a Solution So I solve the system $$cx+10x^2=0$$ $$x-2y=0$$ So if $$x=2y$$ I have $$2yc+10*4y^2=2yc+40y^2=y(2c+40y)=0$$ and I get $$y=0$$ and $$y=-\frac{c}{20}$$ f I...
  24. E

    A Differential Equation to Difference Equation

    Hi all, I am a bit new in this, am trying to learn DE, dynamical systems, & chaos. I am looking into some answers for the following questions: 1) Is it always possible to derive a difference equation for every differential equation, and if so how do we do that? 2) Consider Lorenz system...
  25. F

    Solve the second-order differential equation

    Homework Statement Homework Equations The Attempt at a Solution Can someone check my answer please ?
  26. F

    Solve the differential equation with constant coefficients

    Homework Statement Homework Equations The Attempt at a Solution [/B] Is my answer correct?
  27. W

    Finding Orthogonal Trajectories (differential equations)

    Homework Statement Find Orthogonal Trajectories of ##\frac{x^2}{a}-\frac{y^2}{a-1}=1## Hint Substitute a new independent variable w ##x^2=w## and an new dependent variable z ##y^2=z## Homework EquationsThe Attempt at a Solution substituting ##x## and ##y## I get...
  28. M

    Solving Differential Equation with Frequency Response

    Homework Statement The AC response of an inductor can be modeled by the following differential equation: L \frac{di}{dt} + iR = V Find, using frequency response, the current of the system when the applied voltage V is: V = V_0 \sin(\omega t) Homework EquationsThe Attempt at a Solution In...
  29. Chromatic_Universe

    I Solving a nonlinear first order differential equation

    (a'[t]/a[t])^2 == K*(A + B*a[t]^-6)^1/2} is the equation to be solved for getting the solution of a(t) in terms of time(t). Any ideas on how to solve this problem? Use of Matlab or Mathematica is accepted.
  30. WMDhamnekar

    MHB Solutions of DE System & 2nd Order Differential Equation

    Hello, $\vec{x'}=\small\begin{pmatrix}1&2\\3&2\end{pmatrix}\vec{x}+t\small\begin{pmatrix}2\\-4\end{pmatrix}$ Now i got the solution to this differential equation system as...
  31. S

    Differential equation with power series method

    Homework Statement I need to solve the DE y’ = x^2y using the power series method Homework Equations y = sum(0->inf)Cnx^n y’ = sum(1->inf)nCnx^(n-1) The Attempt at a Solution I plug in the previous two equations into the DE. What is the general procedure for these problems after that...
  32. Hawkingo

    Help in solving an inexact differential equation

    Homework Statement The question is to solve the inexact equation by turning it into exact.the equation is ##( x + y + 4 ) d x + ( - x + y + 6 ) d y = 0## Where "x" and "y" are variable. 2. Homework Equations [/B] 1.(x+y+4)=m and (-x+y+6)=n 2.Integrating Factor =##\frac { 1 } { x ^ { 2 } + y...
  33. Kaguro

    A differential equation with undetermined coefficients

    Homework Statement Solve the following DE with the method of undetermined coefficients: y'' + 4y = 2cos(3x)cos(x) Homework Equations 2cos(3x)cos(x) = cos(4x) + cos(2x) The Attempt at a Solution Let's split the particular integral into two parts: yp1 and yp2. So yp1 is solution for RHS=cos(4x)...
  34. Felipe Lincoln

    First order differential equation

    Homework Statement Solve the following differential equation such that ##x(0)=1##. ## \dfrac{dx}{dt} + 2tx = 3e^{-t^2}+t## Homework Equations Integrating factor: ##\mu(t) = exp\left(\int_0^t2t \right)## The Attempt at a Solution I used the integrating factor and then got the solution ##x(t) =...
  35. N

    1st Order Differential Equation - Power Series Method

    Homework Statement The Attempt at a Solution I have deliberately made several obvious steps, because I keep ending up here. However I have no idea what to do from here. I thought about the fact, that differential equations have the solution ##x = x_{HOM} + x_{Inhom}##, but the ##x_{HOM}##...
  36. Josu Aguirrebeitia

    A Solution form for the following differential equation

    Hi. After arranging the dynamic contact between a elastic ball against a flat, I have reached the following differential equation for the motion during the contact: m·x’’+(k+c·x’)·x^n=0 with m,c,k>0 and for exponent n --> 1<n<2 Any functional form for this equation? I have solved it...
  37. G

    Transmission line: leakage current differential equation

    Homework Statement I have a coaxial cable with internal conductor of radius r1 and external conductor of radii r2 and r3. The material of the conductors has a conductivity ##\sigma_1##. Between the conductors there is a imperfect dielectric of conductivity ##\sigma_2##. Consider the...
  38. K

    Differential equation for the acceleration of an oscillating particle

    Homework Statement acceleration of certain oscillating particle described by a = -x/9 determine the position of this particle when t = 3π/2 if when t=0 x=0 and v=v0 Homework Equations dv/dt=a The Attempt at a Solution frankly I am not sure how to start but i have two ways in my mind(even i...
  39. H

    MHB Help: Differential equation Romeo & Juliet

    Hi, I have to make an assignment on differential equations and Romeo and Juliet. r(t) is romeo's love for Juliet at time t, j(t) is Juliet's love for Romeo at time t So far, it is given: dr/dt=-j and dj/dt=r. It is also given that Romeo & Juliet's families are enemies, thus the initial...
  40. S

    I Y'' + y = 0 solution and recursion relation

    I've found the general solution to be y(x) = C1cos(x) + C2sin(x). I've also found a recursion relation for the equation to be: An+2 = -An / (n+2)(n+1) I now need to show that this recursion relation is equivalent to the general solution. How do I go about doing this? Any help would be...
  41. T

    Find v(t) from Newton's Second Law and Differential Equation

    <Moderator's note: Moved from a technical forum and thus no template.> Is what I have done correct ? I want to find v(t) from Sigma F = m*a. I have gravity force mg pointing downward with positive direction and resistive force R = -b*v^2 pointing upwards with negative direction are acting on a...
  42. RpWinter

    Solving the differential equation of planetary motion

    Hey, this is how i tried solving the differential equation The solution however does not match the general solution of the equation. Also differentiating it twice does not give me the previous equation. Please tell me if i did some mistake while solving. I already know how to solve by finding...
  43. B

    Variation of Parameters to solve a second order ODE

    Homework Statement The question I am working on is the one in the file attached. Homework Equations y = u1y1 + u2y2 : u1'y1 + u2'y2 = 0 u1'y1' + u2'y2' = g(t) The Attempt at a Solution I think I have got part (i) completed, with y1 = e3it and y2 = e-3it. This gives a general solution to the...
  44. I

    I Drude Model Permittivity Formula - e^iwt or e^(-iwt)?

    I've been having a sign problem while deriving the permittivity formula using Drude model, and I found out that the problem came from the fact that complex field vectors are expressed with e-iwt, not eiwt, thus producing (-iwt) term when differentiated...
  45. R

    MHB Differential equation with a matrix

    Suppose we have the matrix $ \mathbf{N} = \begin{bmatrix} 4 & -2 \\ -2 & 1 \end{bmatrix}$ and $\mathbf{X} = \begin{bmatrix}x \\ y \end{bmatrix}$. I want to solve $\displaystyle \frac{d\mathbf{X}}{dt} = \mathbf{NX}$. The eigenvalues of the matrix are $\lambda_1, \lambda_2 = 0,5$ and eigenvectors...
  46. P

    I General solution to a 2nd-order differential equation

    Dear all I've been trying to work out the general solution to a 2nd order ODE of the form f''(x)+p(x)*f(x)=0 p(x) is a polynomial for my case. I believe series method should work, but for some reason I would prefer a general solution using other methods. I'll be much appreciative for any help...
  47. karush

    MHB 2.1.2 Find the general solution of the given differential equation

    Find the general solution of the given differential equation $\displaystyle y^\prime - 2y = t^2 e^{2t}$ Obtain $u(t)$ $\displaystyle u(t)=\exp\int -2 \, dx =e^{-2t}$ Multiply thru with $e^{-2t}$ $e^{-2t}y^\prime + 2e^{-2t}y = t^2 $ Simplify: $(e^{-2t}y)'= t^2$...
  48. C

    Calculations for a capacitor charging from another capacitor

    Hi all, I'd appreciate help in calculating the voltages in the circuit shown. I thought it should be fairly straight forward, but it has me stumped. This is a sample-and-hold circuit for an ADC -- the switch is closed to charge the hold capacitor with the sample voltage and then opened to...
  49. B

    I Differential equation from derivative of time dilation

    Hi all! I was messing around with the equation for time dilation. What I wanted to do was see how the time of a moving observer ##t'## changed with respect to the time of a stationary observer ##t##. So I differentiated the equation for time dilation ##t'## with respect to ##t##: $$\frac {dt'}...
  50. karush

    MHB Is the given function a solution of the differential equation?

    $\textsf{ Verify the following given functions is a solution of the differential equation}\\ \\$ $y''''+4y'''+3y=t\\$ $y_1(t)=t/3$ \begin{align*} (t/3)''''+4(t/3)'''+(t/3)&=t\\ 0+0+t&=t \end{align*} $y_2(t)=e^{-t}+t/3$ \begin{align*} (e^{-t}+t/3)''''+4(e^{-t}+t/3)'''+3(e^{-t}+t/3)&=t\\...
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