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...
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...
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...
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...
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...
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##...
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...
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...
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...
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...
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) =...
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...
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...
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}...
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...
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...
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...
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!
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...
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...
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 -...
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-...
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...
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...
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...
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...
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...
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}...