Ode Definition and 1000 Threads

  1. 1

    Solving an ODE by the method of Integrating Factors

    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...
  2. m4r35n357

    Python Lorenz ODE solver in 35 lines of pure Python

    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...
  3. W

    How to Simplify Nonlinear First-Order ODEs in Physics Problems?

    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...
  4. P

    MHB Can You Solve These Challenging ODE Exam Questions?

    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...
  5. A

    Using Mathematica to solve an ODE

    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}}]
  6. Martin T

    I About Arnold's ODE Book Notation

    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...
  7. B

    ODE question: Understanding a step in the solution

    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...
  8. 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...
  9. B

    Solving a second order ODE using reduction of order

    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...
  10. R

    A Runge-Kutta: Maintaining Units in Numerical Methods

    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 How can scaling symmetry transform differential equations in geometry?

    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. M

    Green's Function for Linear ODE

    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...
  13. C

    Non-linear second-order ODE to Fuchsian equation

    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...
  14. evinda

    MHB Solving the ODE $u_t+u^2 u_x=0$ with Initial Condition $u(x,0)=2+x$

    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...
  15. L

    Troubleshooting First Order ODE Conversions

    What am I doing wrong here in my attachment?
  16. M

    A Inverse ODE, Green's Functions, and series solution

    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...
  17. mertcan

    A Spectral Theorem to Convert PDE into ODE

    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...
  18. incredibe1999

    Maple ODE System Solution Maple: Plotting Earth's Orbit with Sun at Origin

    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...
  19. yecko

    Solving a Linear ODE using a power series

    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...
  20. T

    Fourier Transformation of ODE

    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...
  21. M

    A Solving an ODE Eigenvalue Problem via the Ritz method

    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) =...
  22. M

    A Legendre's ODE: Fundamental Solution for L = 2, 3, 4...

    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...
  23. learn.steadfast

    I Minimizing a non-linear ODE in f, f_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...
  24. Tunalover

    I A common 2nd order ODE from dynamics but....

    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...
  25. A

    I How to find a solution to this linear ODE?

    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...
  26. 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...
  27. yecko

    ODE problem -- Find the amount salt in the tank as water flows through it

    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...
  28. yecko

    Solving Linear ODEs: How to Obtain the Highlighted 1?

    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...
  29. A

    How Do You Model Free Fall with Air Resistance in Differential Equations?

    <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...
  30. C

    MHB Peak in single ODE within a system

    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...
  31. A

    Unraveling an ODE: Solving a First Order Equation with Separation of Variables

    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...
  32. 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}...
  33. 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...
  34. M

    Green's Function of Linear ODE

    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...
  35. A

    I Trying to obtain a 2nd order ODE

    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.
  36. K

    A How Does Green's Function Solve Nonlinear Boundary Value Problems?

    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...
  37. O

    I Using Fourier Transform to Solve ODE with Initial Conditions

    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...
  38. G

    Basic First ODE and Separation problem

    <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...
  39. S

    I Can initial conditions for an ODE be given by functions instead of constants?

    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...
  40. 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...
  41. K

    Second order ODE: finding solution.

    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] -...
  42. K

    B Solve ODE with Fractional Term: Find Solutions

    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 )...
  43. H

    I Equality of two particular solutions of 2nd order linear ODE

    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...
  44. 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!
  45. D

    MATLAB Matlab Sim Code for ODE & Gillespie for Reversible Reaction

    Here is the code you input into matlab. Aini etc, are the initial values of the population densities. A for predator, B for Prey. % example for ODE and Gillespie % one reversible reaction b1 = .033; bo = .00047; a1 = .0022; ao = .00055; Aini = 5168; Bini = 34; %% Basic ODE simulation...
  46. P

    I 2nd order ODE numerical solution

    I would like to solve the following differential equation, it seems easy but only given one initial value. y''(x) = ln(ln(x)) y(5) = 0 Solve for y(10) I know it can be directly integrated but cannot be expressed in terms of elementary functions. Most numerical method involves expressing the...
  47. 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...
  48. S

    I Converting a Single ODE to Matrix Form for Eigenvalue Analysis

    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...
  49. ReidMerrill

    2nd order ODE: modeling a spring

    Suppose a spring with spring constant 6N/m is horizontal and has one end attached to the wall and the other end attached to a 3 kg mass. Suppose the friction/damping constant is 1 N s/m Set up a differential equation that describes this system with x denoting displacement of the mass from...
  50. Another

    Problem ODE from mechanics equation

    The particles are moving under force F = rK when r = 1/u and u = 1/r I tried to solve the problem by defining a variable v = u4 and u = v1/2 But I can not divide variable v out of variable u I want to find r (Radius) of orbit
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