# ode

1. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...

23. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### (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}...