Ode Definition and 1000 Threads

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...

Hi PF! Given the ODE $$f'' = -\lambda f : f(0)=f(1)=0$$ we know ##f_n = \sin (n\pi x), \lambda_n = (n\pi)^2##. Estimating eigenvalues via Rayleigh quotient implies $$\lambda_n \leq R_n \equiv -\frac{(\phi''_n,\phi_n)}{(\phi_n,\phi_n)}$$ where ##\phi_n## are the trial functions. Does the...

1. y' + y = x y2/32. The problem states we need to solve this ODE by using the method of integrating factors. Every example I found on the internet involving this method was of the form: y' + Py = Q Where P and Q are functions of x only. In the problem I was given however, Q is a function of...

Features: No external dependencies. Arbitrary order simulation (accuracy limited by float precision). No finite difference errors. Can be extended to arbitrary precision with gmpy2 and two more lines. Enjoy! Parameters: order step size number of steps initial conditions (x, y, z) parameters...

Homework Statement Let $$\frac{1}{2}\dot{r}^2=e+\frac{m}{r}-\frac{L^2}{2r^2}$$ where L is angular moment, and e is energy (so I guess I'll take as constants for now...) Homework Equations Not sure for now. The Attempt at a Solution So, if I let $$u=\frac{1}{r}$$ then my equation becomes...

Hi, I need to do these for an exam and I can't find any way to solve them with what my professor has taught. If you can help me answer even just one of them I'd be very grateful! 1. Which of the differential equations is/are non linear? a. Only I b. Only II c. Only I and III d. Only II and IV...

Homework Statement Homework Equations The Attempt at a Solution I used the NDSolve function from mathematic but its giving me problems. What is the correct way to enter the equation?[/B] soln = NDSolve[{y''[t] = (-9.8/5)*sin (t), y[0] = 20, y, {x, 0, 12}}]

In Arnold's book, ordinary differential equations 3rd. WHY Arnold say Tg:M→M instead of Tg:G→S(M) for transformations Tfg=Tf Tg, Tg^-1=(Tg)^-1. Let M be a group and M a set. We say that an action of the group G on the set M is defined if to each element g of G there corresponds a...

Homework Statement Hi there, I don't nee help with solving a question, so much as understanding a step in the provided worked solution. It's using variation of parameters to solve the ode y''+ y = g(t). I've attached the steps in the picture file, and the bit after the word 'Now' what are they...

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...

Homework Statement Hi there, I have an assignment which involves using reduction of order to solve for a second solution to an ode (the one attached). However this is a method I am new to, and though I have tried several times, I'm somehow getting something wrong because the LHS and RHS are not...

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) =...

Homework Statement Find the Green's function for $$f''(x) + \cos^2 a f(x) = 0;\\ \pm f'(x) + \cos a \cot a f(x)|_{x=x_0(a)}=0$$ where ##a## is a parameter and ##x_0## is defined as $$x_0(a) = \sec a\arcsin(\cos a)$$. Homework Equations Standard variation of parameters The Attempt at a...

Homework Statement z\frac{d^2z}{dw^2}+\left(\frac{dz}{dw}\right)^2+\frac{\left(2w^2-1\right)}{w^3}z\frac{dz}{dw}+\frac{z^2}{2w^4}=0 (a) Use z=\sqrt y to linearize the equation. (b) Use t=\frac{1}{w} to make singularities regular. (c) Solve the equation. (d) Is the last equation obtained a...

Hello! (Wave) I want to solve the equation $u_t+u^2u_x=0$ with $u(x,0)=2+x$. I have tried the following: The characteristic curves for $u_t+u^2 u_x=0$ are the solutions of the ode $\frac{dx}{dt}=u^2$. We have that $\frac{d}{dt}u(x(t),t)=0$, implying that $u(x(t),t)=c$. The characteristic...

What am I doing wrong here in my attachment?

Hi PF! One way to solve a simple eigenvalue problem like $$y''(x)+\lambda y(x) = 0,\\ y(0)=y(1)=0$$ (I realize the solution's amplitude can be however large, but my point here is not to focus on that) is to solve the inverse problem. If we say ##A[u(x)] \equiv d^2_x u(x)## and ##B[u(x)] \equiv...

Hi, in the link https://math.stackexchange.com/questions/1465629/numerically-solving-a-non-linear-pde-by-an-ode-on-the-fourier-coefficients there is a nice example related to spectral theorem using Fourier series. Also in the link...

Hi! \begin{cases} \dot{q} = a \left( 1 - q^2 \right) \\ \dot{a} = - \alpha - a^2 q\end{cases} \qquad \alpha \in (0, 1 ) I've looked into this ODE system about 7 months now, but I've not got anything promising how to write down the solution. I'm mostly interested in q-serie. (To those of you...

Hey there! I was trying to plot a ODE solution, but am not getting what I should be. It is actually plotting the orbit of Earth with Sun at the origin. My equations The (-0.00011847) is GM. The Initial Conditions: The plot I get: Should not I be getting a elliptic/circular plot as the...

Homework Statement Homework Equations Power series ODE The Attempt at a Solution [/B] Sorry for not typing all those things out from my phone.. How can I get C1? And how can I put the solution in the required format? (I don't know how to put it in summation sign... and i cannot even solve...

Find Green's function of $$K(f(x)) = (1-x^2)f''(x)-2xf'(x)+\left(2-\frac{1}{1-x^2}\right)f(x):x\in[cos(\alpha),1]$$ subject to boundary conditions: $$f|_{x=1} < \infty\\ f|_{x=\cos(\alpha)} = 0.$$ Two fundamental solutions are associated Legendre polynomials (after all, this is Legendre's...

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: Homework Equations a) The Fourier Transform b) The Inverse Fourier Transform The Attempt at a Solution I started by applying the Fourier...

Hi PF! I want to solve ##u''(x) = -\lambda u(x) : u(0)=u(1)=0##. I know solutions are ##u(x) = \sin(\sqrt{\lambda} x):\lambda = (n\pi)^2##. I'm trying to solve via the Ritz method. Here's what I have: define ##A(u)\equiv d^2_x u## and ##B(u)\equiv u##. Then in operator form we have ##A(u) =...

Hi PF! I'm wondering what the fundamental solution is for this ODE $$ f''(x)+\cot (x) f'(x) + \left( 2-\frac{L^2}{\sin(x)} \right) f(x) = 0 : L = 2,3,4... $$ I know one solution is $$ (\cos(x)+L)\left(\frac{1-\cos(s)}{1+\cos(s)}\right)^{L/2} $$ but I don't know the other. Mathematica isn't...

I'm working on a physics "potential" problem and trying to create an alternate function to describe the potential energy. I'm having trouble figuring out how to solve a nonlinear ODE, or even a limiting boundary for minimizing a drop off shape function. I was able to reduce my problem to the...

Consider a simple single degree-of-freedom (SDOF) spring-mass-dashpot dynamic system with spring rate k, mass m, and viscous damping coefficient c. Dimension x is the absolute displacement of the mass. The base input translation is y. A dot notation indicates differentiation with respect to...

I want to find solution to following ODE $$ \frac{d \bar h}{dt} + \frac{K}{S_s} \alpha^2 \bar h = -\frac{K}{S_s} \alpha H h_b(t) $$ I have solved it with integrating factor method with ## I=\exp^{\int \frac{1}{D} \alpha^2 dt} ## as integrating factor and ##\frac{K}{S_s} = \frac{1}{D} ## I have...

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...

During solution of a PDE I came across following ODE ## \frac{d \bar h}{dt} + \frac{K}{S_s} \alpha^2 \bar h = -\frac{K}{S_s} \alpha H h_b(t) ## I have to solve this ODE which I have done using integrating factor using following steps taking integrating factor I=\exp^{\int \frac{1}{D} \alpha^2...

Homework Statement A tank originally contains 100 gal of fresh water. Then water containing 1/2 lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After 10 min the process is stopped, and fresh water is poured into the...

Homework Statement How to obtain the "1" highlighted? Homework Equations multiply by μ then by dt (integration )to both sides The Attempt at a Solution [/B] lets just consider part "2y/t": ∫2y/t from pi/2 to t =ln(t^2/(pi^2/4)) from pi/2 to t =ln1-ln(t^2/(pi^2/4)) = -ln(t^2/(pi^2/4)) how...

<Moderator's note: Moved from a technical forum and thus no template.> An object with mass 96 kg is given an initial downward velocity −3m/s in a medium that exerts a resistive force with magnitude proportional to the square of the speed. The resistance is 60 N when the velocity is −2m/s. Use...

Hi all, I have the system of nonlinear ODEs: $$ \begin{align} \frac{dX}{dt}=&-k_+ A X+k_-Y \\ \frac{dY}{dt}=&\ k_+ A X-k_-Y-\alpha k_+ X Y +\beta Z \\ \frac{dZ}{dt}=&\ \alpha k_+ X Y -\beta Z \end{align} $$ I also have a conservation law that says $D=X+Y+2Z$. Obviously it is not possible to...

Homework Statement Solve: ##\frac {dx} {dt}## ##\text{= 8-3x , x(0)=4}## The Attempt at a Solution Step 1: ##\int \frac 1 {8-3x} \, dx## = ##\int \, dt## Step 2: - ##\frac 1 3## ##\text{ln|8-3x| = t+c}## From here I am going to try to get it into explicit form Step 3...

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...

Homework Statement Find Green's function of ##u''+u=f##. Homework Equations What we all know. The Attempt at a Solution Let Greens function be ##G##. Then ##G''+G=\delta(x-x_0)##. This admits solutions superimposed of sine and cosine. Let's split the function at ##x=x_0##. Then we require...

Hi I'm having a slight issue trying to obtain a 2nd order ODE with respect to x (so involves implicit differentiation in this case) from the equation below. I would greatly appreciate any help or tips to solve this problem. I've removed the coefficients to make things a litter easier. Thank you.

I have this BVP $$u''+u' =f(x)-\lambda |u(x)| $$, ##x\in [0,1]## we BC ## u(0)=u(1)=0##. Following an ''algorithm'' for calculating the green's function I got something like $$g(x,t)=\Theta(x-t)(1+e^{t-x}) + \frac{e^{t}-e}{e-1} +\frac{e-e^{t}}{e-1}e^{-x}$$. At some point there is this integral...

Hi, let's take this ode: y''(t) = f(t),y(0)=0, y'(0)=0. using the FT it becomes: -w^2 Y(w) = F(w) Y(w)=( -1/w^2 )F(w) so i can say that -1/w^2 is the Fourier transorm of the green's function(let's call it G(w)). then y(t) = g(t) * f(t) where g(t) = F^-1 (G(w)) (inverse Fourier transorm) how can...

<Moderator's note: Moved from a technical forum and thus no template.> Hi all, hope you all had a great Christmas, I had a difficulty with theses two problems, 1. Show that y=1/x + x/2 is a solution of the differential equation, dy/dx=1-y/x, where y=1.5 and x=2 2. Solve the differential...

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...

Homework Statement d2u/d2x + 1/2Lu = 0 where L is function of x Homework Equations I am try to find solutions y1 and y2 of this equation. The Attempt at a Solution y = [cos √(L/2) x] + [sin √(L/2) x] y' = - [√(L/2) sin √(L/2) x] + [ √(L/2) cos √(L/2) x] y'' = -[(L/2) cos √(L/2) x] -...

Hi, This is equation I need to find solutions for d2u/d2x + 1/2Lu = 0 where L(x) I understand we can remove fraction from second term. 2 [d2 u/d2x ] + Lu = 0 now how do I find solution of this equation ? How do we deal with L ? because usually we have Y'(dy/dx or in this case du/dx )...

I got the following two integral for the a particular solution of a 2nd order linear ODE $$(D-a)(D-b)y = g(x)$$ by using inverse operators ##\frac{1}{D-a}## and ##\frac{1}{D-b}##. The two different integrals are obtained by operating these operators in different order on y to get a particular...

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!

This is the code for Sensitivity Analysis via Rosenwasser's method. Code was for my Masters Thesis, so maybe it will be useful to someone in Dynamical Systems or Modeling with ODE's function ode45_both_age %-------------------------------------------------------------------------- % Solves...