ode

  1. Richard Parker

    I Difference between stationary and steady state

    I was recently working on a problem of Griffiths and in the solution's manual it used an argument to solve a diffential equation that caught my attention. It said that it would look first to the steady state solution of the ODE. I tought "All right, I get that" but when I got to translate the...
  2. m4r35n357

    I How bad is my maths? (numerical ODE method)

    I have written some ODE solvers, using a method which may not be well known to many. This is my attempt to explain my implementation of the method as simply as possible, but I would appreciate review and corrections. At various points the text mentions Taylor Series recurrences, which I only...
  3. DeclanKerr

    RLC Circuit Analysis with system of ODEs

    Summary: Looking for guidance on how to model an RLC circuit with a system of ODES, where the variables are the resistor and inductor voltages. This is a maths problem I have to complete for homework. The problem is trying to prove that the attached circuit diagram can be modeled using the...
  4. m4r35n357

    Python Damped & Driven Pendulums (in _pure_ Python)

    This is another application of using Taylor recurrences (open access) to solve ODEs to arbitrarily high order (e.g. 10th order in the example invocation). It illustrates use of trigonometric recurrences, rather than the product recurrences in my earlier Lorenz ODE posts. Enjoy! #!/usr/bin/env...
  5. T

    A Implicit Euler method with adaptive time step and step doubling

    For Initial Value problems I want to implement an ODE solver for implicit Euler method with adaptive time step and use step doubling to estimate error. I have found some reading stuff about adaptive time step and error estimation using step doubling but those are mostly related to RK methods. I...
  6. Danny Boy

    Approximate solutions to Kuramoto synchronization model

    According to the wiki entry 'Kuramoto Model', if we consider the ##N=2## case then the governing equations are $$\frac{d \theta_1}{dt} = \omega_i + \frac{K}{2}\sin(\theta_2 - \theta_1)~~~\text{and}~~~\frac{d \theta_2}{dt} = \omega_i + \frac{K}{2}\sin(\theta_1 - \theta_2),$$ where ##\theta_i##...
  7. T

    I Nonlinear Second Order ODE

    I'm trying to solve the following nonlinear second order ODE where ##a## and ##b## are constants: $$\frac{d^2y}{dx^2}+\frac{1}{x}\frac{dy}{dx}-\frac{y}{ay+b}=0$$ It looks somewhat like the modified Bessel equation, except the third term on the left makes it nonlinear. I've been trying to...
  8. H

    I How to solve an ODE to find its solution

    Salutations, I have a problem when I approach this ODE: $$\left(\frac{y}{y'}\right)^2+y^2=b^2\left(x-\frac{y}{y'}\right)^2$$ I have done a series of steps as I show in this link: https://drive.google.com/file/d/1Ht4xxUlm7vXqg4S5-wirKwm7vTESU3mU/view?usp=sharing But I'm not convinced that those...
  9. 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...
  10. R

    A Runge Kutta - units

    When conducting numerical methods using 4th Order Runge-Kutta do the physical units have to be maintained? This never occurred to me until I was writing out all the steps in detail when showing someone I work with the method using a simple projectile motion with drag. It had 4th Order time...
  11. C

    A ODE from Geometry Problem

    A geometry problem I'm working on has boiled down to finding a function ##f(t)## such that $$f'' + \frac{2}{t}f' + \frac{f'^2}{\left( 1 - \frac{f}{t} \right) t } + \frac{f'f}{\left(1- \frac{f}{t} \right) t^2} = 0$$ It has two fairly simple solutions, namely ##f(t) = a## and ##f(t) =...
  12. T

    Fourier Transformation of ODE

    1. Homework Statement I am to solve an ODE using the Fourier Transform, however I am quite inexperienced in using this method so I'd like some advice: 2. Homework Equations a) The Fourier Transform b) The Inverse Fourier Transform 3. The Attempt at a Solution I started by applying...
  13. SemM

    A Solve a non-linear ODE of third order

    Hi, I tried to solve the following in Wolfram alpha: y''' + (1-x^2)y=0 y(0)=0 y'(0)=0 y''(0)=0 however, I got answer which cannot be reproduced (even at wolfram pages). I have tried ODE45 in MATLAB, but it only gives a plot. Is there any way to solve this analytically or numerically to give...
  14. SemM

    A Non-selfadjointness and solutions

    Hi, I have the two operators: \begin{equation} Q = i\hbar \frac{d}{dx} - \gamma \end{equation} \begin{equation} Q' = -i\hbar \frac{d}{dx} - \gamma \end{equation} where ##\gamma## is a constant. Both of these are not self-adjoint, as they do not follow the condition: \begin{equation}...
  15. S

    Pressure trace of a tank fed by a compressor

    G'Day All, This is my first post so please let me know if I have completed this form incorrectly, or missed a point of etiquette etc... 1. Homework Statement The problem is to determine the pressurisation rate of a tank being filled by a pipe connected to a compressor. Assumptions: Pipe...
  16. S

    I Initial conditions for an ODE

    Hi, I am trying to solve an ODE, however, the initial conditions are not known. From PDE examples, which are quite different, I see that some examples have initial conditions given by functions, and not by constants, i.e:: y(0) = x^2 I may have not modeled the problem correctly yet, however, I...
  17. S

    I Consequences on a system of ODEs after performing operations

    Hi, I have derived a matrix from a system of ODE, and the matrix looked pretty bad at first. Then recently, I tried the Gauss elimination, followed by the exponential application on the matrix (e^[A]) and after another Gauss elimination, it turned "down" to the Identity matrix. This is awfully...
  18. S

    I Convert complex ODE to matrix form

    Hi, I have the following complex ODE: aY'' + ibY' = 0 and thought that it could be written as: [a, ib; -1, 1] Then the determinant of this matrix would give the form a + ib = 0 Is this correct and logically sound? Thanks!
  19. S

    I How to study an ODE in matrix form in a Hilbert space?

    Hello, I have derived the matrix form of one ODE, and found a complex matrix, whose phase portrait is a spiral source. The matrix indicates further that the ODE has diffeomorphic flow and requires stringent initial conditions. I have thought about including limits for the matrix, however the...
  20. S

    I Convert an ODE to matrix form

    Hi, I have the following ODE: aY'' + bY' + c = 0 I would like to convert it to a matrix, so to evaluate its eigenvalues and eigenvectors. I have done so for phase.plane system before, however there were two ODEs there. In this case, there is only one, so how does this look like in a matrix...
  21. C

    Second order(?) ODE + Runge-Kutta method question

    1. Homework Statement When a rocket launches, it burns fuel at a constant rate of (kg/s) as it accelerates, maintaining a constant thrust of T. The weight of the rocket, including fuel is 1200 kg (including 900 kg of fuel). So, the mass of the rocket changes as it accelerates: m(t) = 1200 -...
  22. B

    Linear ordinary differential equation.

    1. Homework Statement ##\dfrac{dy}{dx} + y = f(x)## ##f(x) = \begin{cases} 2 \qquad x \in [0, 1) \\ 0 \qquad x \ge 1 \end{cases}## ##y(0) = 0## 2. Homework Equations 3. The Attempt at a Solution Integrating factor is ##e^x## ##e^x\dfrac{dy}{dx} + e^x y = e^x f(x)## ##\displaystyle...
  23. B

    A simple DE.

    1. Homework Statement ##(2x + 3y + 1)dx + (4x + 6y + 1) dy = 0## ##y(-2) = 2## 2. Homework Equations 3. The Attempt at a Solution Let ##z = 2x + 3y## then ##z^\prime = 2 + 3y^prime## ##\displaystyle \dfrac{(z + 1)}{2z + 1} + \dfrac13\left({dz \over dx} - 2\right) = 0## ##\dfrac{dz}{dx}...
  24. S

    I Constructing a 2nd order homogenous DE given fundamental solution

    1. Homework Statement Given a set of fundamental solutions {ex*sinx*cosx, ex*cos(2x)} 2. Homework Equations y''+p(x)y'+q(x)=0 det W(y1,y2) =Ce-∫p(x)dx 3. The Attempt at a Solution I took the determinant of the matrix to get e2x[cos(2x)cosxsinx-2sin(2x)sinxcosx-cos(2x)sinxcosx-...
  25. whatisgoingon

    Second order ODE into a system of first order ODEs

    1. Homework Statement The harmonic oscillator's equation of motion is: x'' + 2βx' + ω02x = f with the forcing of the form f(t) = f0sin(ωt) 3. The Attempt at a Solution So I got: X1 = x X1' = x' = X2 X2 = x' X2' = x'' ∴ X2' = -2βX2 - ω02X1 + sin(ωt) The function f(t) is making me doubt...
  26. Poetria

    Solutions space (ODE)

    1. Homework Statement How many functions y(t) satisfy both y''+t^2*y=0 and y(0)=6? 2. The attempt at a solution As this is a second order differential equation, two initial conditions (for y and y') would be needed to obtain a unique solution (cf. existence and uniqueness theorem). So the...
  27. M

    Properties of Solutions of Matrix ODEs

    1. Homework Statement We assume from ODE theory that given a smooth A: I → gl(n;R) there exists a unique smooth solution F : I → gl(n;R), defined on the same interval I on which A is defined, of the initial value problem F' = FA and F(t0) = F0 ∈ gl(n;R) given. (i) Show that two solutions Fi...
  28. M

    Proving basic linear ODE results

    1. Homework Statement Please bear with the length of this post, I'm taking it one step at a time starting with i) Let A: I → gl(n, R) be a smooth function where I ⊂ R is an interval and gl(n, R) denotes the vector space of all n × n matrices. (i) If F : I → gl(n, R) satisfies the matrix ODE...
  29. Sirsh

    4DOF Spur Gear System - Eigenvalues not corresponding with EQs?

    Hi there, I am modelling a four degree of freedom system which is the dynamics of two spur gears in mesh, having two rotational and two translation degrees of freedom, respectively, a diagram exhibiting the system can be seen below. I have derived the equations of motion (EOM) and...
  30. Euler2718

    (Ordinary) Differential Equation Trouble

    1. Homework Statement Find the solution of the differential equation by using appropriate method: t^{2}y^{\prime} + 2ty - y^{3} = 0 2. Homework Equations I'm thinking substitution method of a Bernoulli equation: v = y^{1-n} 3. The Attempt at a Solution t^{2}y^{\prime} + 2ty - y^{3}...
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