What is Ode: Definition and 1000 Discussions

An ode (from Ancient Greek: ᾠδή, romanized: ōdḗ) is a type of lyrical stanza. It is an elaborately structured poem praising or glorifying an event or individual, describing nature intellectually as well as emotionally. A classic ode is structured in three major parts: the strophe, the antistrophe, and the epode. Different forms such as the homostrophic ode and the irregular ode also enter.
Greek odes were originally poetic pieces performed with musical accompaniment. As time passed on, they gradually became known as personal lyrical compositions whether sung (with or without musical instruments) or merely recited (always with accompaniment). The primary instruments used were the aulos and the lyre (the latter was the most revered instrument to the ancient Greeks).
There are three typical forms of odes: the Pindaric, Horatian, and irregular. Pindaric odes follow the form and style of Pindar. Horatian odes follow conventions of Horace; the odes of Horace deliberately imitated the Greek lyricists such as Alcaeus and Anacreon. Irregular odes use rhyme, but not the three-part form of the Pindaric ode, nor the two- or four-line stanza of the Horatian ode. The ode is a lyric poem. It conveys exalted and inspired emotions. It is a lyric in an elaborate form, expressed in a language that is imaginative, dignified and sincere. Like the lyric, an ode is of Greek origin.

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  1. SchroedingersLion

    A Question about this Separable ODE statement in a book

    Greetings, I have a question to the following section of the book https://www.springer.com/gp/book/9783319163741: I understand that the equation is separable, since I can just write $$ \int_{x_0}^{x} \frac {1}{V(x', \xi, \eta)}dx' =\int_{0}^{t}dt' .$$ However, without knowing the exact shape...
  2. BlueTempus

    ODE -> Transfer Function Assistance

    Homework Statement:: ODE -> Transfer Function Assistance Relevant Equations:: Newtonian physics, buoyancy, drag [Mentor Note -- thread moved to DE from the schoolwork forums, since it is for work and not schoolwork] Hello all, I'm new here but I'm looking for a bit of guidance with a...
  3. G

    Method of characteristics: Discontinuous source

    Hello all, this question really has me and some friends stomped so advice would be appreciated. Ok so, the relevant (dimensionless) continuity equation I have found to be $$\frac{\partial\rho}{\partial t} + (1-2\rho)\frac{\partial \rho}{\partial x} = \begin{cases} \beta, \hspace{3mm} x < 0 \\...
  4. karush

    MHB -b.2.2.32 First order homogeneous ODE

    \[ \dfrac{dy}{dx} =\dfrac{x^2+3y^2}{2xy} =\dfrac{x^2}{2xy}+\dfrac{3y^2}{2xy} =\dfrac{x}{2y}+\dfrac{3y}{2x}\] ok not sure if this is the best first steip,,,, if so then do a $u=\dfrac{x}{y}$ ?
  5. karush

    MHB -2.2.31 First order homogeneous ODE

    I OK going to do #31 if others new OPs I went over the examples but? well we can't 6seem to start by a simple separation I think direction fields can be derived with desmos
  6. Avatrin

    A Numerically solving matrix Riccati ODE

    Hi I need to solve an equation of the form $$\dot{X}(t) = FX(t) + X(t)F^T + B$$ All of these are matrices. I have an initial condition X(0)=X_0. However, I have no idea how to proceed. How can I make any progress?
  7. Avatrin

    A Numerical Solution to Random Linear Non-Homogeneous ODE

    Hi I am trying to learn optimal estimation by reading Gelbs Applied Optimal Estimation, and I am having hard time with finding \Gamma defined as the following: $$ \Gamma_k w_k = \int_{t_k}^{t_{k+1}} e^{F(t_{k+1} - \sigma)} G(\sigma) w(\sigma) d\sigma$$ Here F is a known matrix. So is G, and w...
  8. chwala

    Understanding the Frobenius Method for Solving Second Order ODEs

    let ##y= \sum_{k=-∞}^\infty a_kz^{k+c}## ##y'=\sum_{k=-∞}^\infty (k+c)a_kz^{k+c-1}## ##y"=\sum_{k=-∞}^\infty (k+c)(k+c-1)a_kz^{k+c-2}## therefore, ##y"+y'\frac {1}{z}+y[\frac {z^2-n^2}{z^2}]=0## =##[\sum_{k=-∞}^\infty [(k+c)^2-n^2)]a_k + a_k-2]z^{k+c} ## it follows that...
  9. PainterGuy

    Solving an ODE with the Laplace transform

    Hi again, The previous problem was done using y′′(t)+2y′(t)+10y(t)=10 with with intial condition y(0⁻)=0. In the following case, I'm using an initial condition and setting the right hand side equal to zero. Find y(t) for the following differential equation with intial condition y(0⁻)=4...
  10. michii15

    I Find the only periodic solution of an ODE

    Find the only periodic solution for 𝑦′+𝑦=𝑏(𝑥) with 𝑏:ℝ→ℝ has a period of 2𝑇 and is 1 for 𝑥(0,𝑇) and −1 for 𝑥(−𝑇,0). The ODE is easy to solve: 𝑦(𝑥)=exp(−𝑥)⋅𝑐+1 and 𝑦(𝑥)=exp(−𝑥)⋅𝑐−1. But how can I find the 𝑐 such that the solution is periodic with a period of 2𝑇? The solution is...
  11. A

    MHB What Method Solves the ODE y''(x) + y'(x) + F(x) = 0?

    y''(x)+y'(x)+F(x)=0 Pleas me a idea
  12. jk22

    I Can Non-Separable ODEs Be Solved with Coordinate Transformations?

    I fell upon such an equation : $$-E'(v)a(1+\frac{cE(v)}{\sqrt{E(v)^2-1}})=vE(v)+c\sqrt{E(v)^2-1}$$ It's not separable in E on one side and v expression on the other. So I'm looking for methods to solve this maybe changes of coordinates ?
  13. Kaguro

    I Solving an ODE with power series

    I have an ODE: (x-1)y'' + (3x-1)y' + y = 0 I need to find the solution about x=0. Since this is an ordinary point, I can use the regular power series solution. Let y = ## \sum_{r=0}^\infty a_r x^r ## after finding the derivatives and putting in the ODE, I have: ## \sum_{r=0}^\infty a_r...
  14. Mentz114

    I ODE Solution Question: Comparing A1(r,ε) and A2(r) with ε=0

    I've been solving these two ODEs ##\frac{d}{d\,r}\,A=F(A,r) + \epsilon f(r)## and ##\frac{d}{d\,r}\,A=F(A,r)##. If the solutions are respectively ##A_1(r,\epsilon)## and ##A_2(r)## then will ##A_1(r,0) = A_2(r)## ? I realize the answer could depend on the actual functions but with the ones...
  15. CrosisBH

    I Solving an ODE with Legendre Polynomials

    From Griffiths E&M 4th edition. He went over solving a PDE using separation of variables. It got to this ODE \frac{d}{d\theta}\left(\sin\theta\frac{d\Theta}{d\theta}\right)= -l(l+1)\sin \theta \Theta Griffths states that this ODE has the solution \Theta = P_l(\cos\theta) Where $$P_l =...
  16. kepherax

    Cauchy-Euler with x=e^t? Differential Equations (ODE)

    I'm fine with this up to a certain point, but I'm not certain if I'm using the substitution correctly. After finding the homogeneous solution do I plug in x= e^t in the original equation and then divide by e^2t to put it in standard form before applying variation of parameters so f=1, or do I...
  17. M

    I ODE Fail: Numerical Solution Oscillations - Possible Solutions | Hi PF

    Hi PF The following ODE $$\ddot x + x - x^3 = 0\\ x(0)=0,\,\,\,\dot x(0) = \frac {1}{ \sqrt 2}$$ is solve exactly with ##\tanh (t/\sqrt 2)##. However, when I try to solve this with MATLAB ode45 (ode23t looks identical) or Mathematica NDSolve I get an oscillatory numerical solution (see...
  18. D

    Structure of code for ODE solution in Octave

    The problem of interest at the moment is the solution of a simple damped oscillator problem, xddot+2*zeta*wn*xdot+wn^2*x = 0 Everything seems to work fine provided I include the values of zeta and wn inside the function that defines the derivative values. But zeta and wn are actually calculated...
  19. D

    Difficulties solving ODE in Octave

    I'm new to using Octave 5.1.0, and a bit confused about how to solve ODEs with Octave. Let me show you a bit of code that I grabbed off a university web site: >> function xdot = pend(x,t) % pend.m xdot(1) = x(2); xdot(2) = - x(1) - 0.1*x(2); end >> sol=lsode( "pend",[0.1, 0.2], t =...
  20. M

    A Signal Processing and the Duffing ODE

    Hi PF! Attached are two plots: signal.pdf is a solution from the Duffing ODE, and plots vertical displacement over time, both the raw signal (blue) and the reconstructed signal from an FFT (red). I've also shown a zoomed in view so you can see how oscillatory the signal is. pow.pdf plots the...
  21. 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...
  22. Luke Tan

    I Can the ODE \psi''-y^2\psi=0 be solved using a general method?

    When reading through Shankar's Principles of Quantum Mechanics, I came across this ODE \psi''-y^2\psi=0 solved in the limit where y tends to infinity. I have tried separating variables and attempted to use an integrating factor to solve this in the general case before taking the limit, but...
  23. WMDhamnekar

    MHB Solving 1st order non-linear ODE

    I want to solve $\d{y}{x}=\frac{3*(2x-7y)+6}{2*(2x-7y)-3}.$ I don't know its step by step solution. But using some trick of solving ordinary differential equation (which I saw on the Internet), I got the following solution:- $-\frac{17}{21}*(3x-2y)+ln(119y-34x-48)=C$. Now how to solve this...
  24. Robin04

    Perturbation theory for solving a second-order ODE

    I have to solve the equation above. I haven't heard about an exact method so I tried to apply perturbation theory. I don't know much about it so I would like to ask for some help. First I put an ##\epsilon## in the coefficient of the non-linear ##\xi^2(t)## term: ##\ddot{\xi}(t)=-b\xi...
  25. R

    I Does this ODE have any real solutions?

    The ODE is: \begin{equation} (y'(x)^2 - z'(x)^2) + 2m^2( y(x)^2 - z(x)^2) = 0 \end{equation} Where y(x) and z(x) are real unknown functions of x, m is a constant. I believe there are complex solutions, as well as the trivial case z(x) = y(x) = 0 , but I cannot find any real solutions. Are...
  26. karush

    MHB -m30 - 2nd order linear homogeneous ODE solve using Wronskian

    2000 Convert the differential equation $$\displaystyle y^{\prime\prime} + 5y^\prime + 6y =0$$ ok I presume this means to find a general solution so $$\lambda^2+5\lambda+6=(\lambda+3)(\lambda+2)=0$$ then the roots are $$-3,-2$$ thus solutions $$e^{-3x},e^{-2x}$$ ok I think the Wronskain...
  27. Robin04

    How is Mathematica solving this ODE with periodic coefficients?

    Mathematica gives this solution but how does it calculate it? What's the method here?
  28. Physics345

    Differential Equation ODE Solution help.

    dM/dY = x+2y+1 dN/dx = 1 (My-Nx)/n = 1 Integrating Factor => e^∫1dx= e^x (xye^x+ye^x+ye^x)dx + (xe^x+2ye^x)dy = 0 dM/dY =xye^x+e^x+2ye^x dN/dx = xye^x+e^x+2ye^x Exact ∫dF/dy * dy = ∫ (xe^x+2ye^x)dy F = xy*e^x + y^2*e^x + c(x) dF/dx = xy*e^x + y*e^x + y^2 * e^x + c'(x)...
  29. J

    I What is the proper format for solving this ODE using an Excel add-in calculator?

    I am attempting to solve an ODE using a Calculus add-in for Excel. I am an industry professional and I have not even thought about Differential Equations in 8 years. The equation that I am attempting to solve is in the form: (1) The ODE solver that I am using solves equations of the form...
  30. karush

    MHB -a.3.2.96 Convert a 2nd order homogeneous ODE into a system of first order ODEs

    given the differential equation $\quad y''+5y'+6y=0$ (a)convert into a system of first order (homogeneous) differential equation (b)solve the system. ok just look at an example the first step would be $\quad u=y'$ then $\quad u'+5u+6=0$ so far perhaps?
  31. 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...
  32. 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...
  33. m4r35n357

    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...
  34. A

    Solving 2nd order ODE in order to get equation for Orbital Trajectory

    I want to solve ##\frac{du^2}{d\theta ^2}+u=\frac{GM}{h^2}## for ##u(\theta)##, where ##\frac{GM}{h^2}=constant##. The given equation is a nonhomogeneous second order linear DE. I begin by solving the associated homogeneous DE with constant coefficients: ##\frac{du^2}{d\theta ^2}+u=0## which...
  35. 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...
  36. I

    I What is a symmetric ODE / what does it mean when an ODE is symmetric?

    How can an ODE be symmetric? How would you plot an ODE to show off this property? (i.e. what would be the axes?)
  37. 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##...
  38. R

    Finding the ODE that describes this circuit + find its transfer function

    As you can see, I've tried using KCL at node A to find the 2nd order ODE that describes this circuit in terms of the capacitor voltage. The problem I run into, however, is that I can't find anything to put the node voltage at A in terms of. I've tried (not shown here) doing mesh current as well...
  39. M

    A How can the stability of an ODE system be determined without solving it?

    Hi PF! Given the ODE system ##x'(t) = A(t) x(t)## where ##x## is a vector and ##A## a square matrix periodic, so that ##A(t) = A(T+t)##, would the following be a good way to solve the system's stability: fix ##t^*##. Then $$ \int \frac{1}{x} \, dx = \int A(t^*) \, dt \implies\\ x(t) =...
  40. Robin04

    Mathematica Plotting the solution of an ODE

    I'm trying to plot the solution to an ODE (with given initial values) but there are some constants in it that I want to evaluate with sliders and I'm not sure what is the right syntax for this. Manipulate[Plot[solution1[t], {t, 0, 10}, PlotRange -> {-Pi, Pi}], {{a, 1, "Driving amplitude"}, 0...
  41. T

    I Nonlinear Second Order ODE: Can We Find an Analytical Solution?

    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...
  42. Z

    Trouble solving an ODE and plotting its phase portrait

    Mentor note: Moved from non-homework forum to here hence no template. So I was able to solve part 1.A of the first problem by hand, the phase portrait is a sideways parabola. However, I want to also show on this on mathematica. I want to solve the equation first and then plot the phase...
  43. chwala

    Solving a first order ODE using the Adomian Decomposition method

    Homework Statement how do we solve the ode ## y'+y^2=-2, y(0)=0## using adomian decomposition method?Homework EquationsThe Attempt at a Solution ##Ly = -2-y^2## ## y= 0 + L^{-1}[-2-y^2]## ##y_{0}= -2t## ##y_{1}= -L^{-1}[4t^2] = -4t^3/3## are my steps correct so far in trying to get the Adomian...
  44. J

    Comp Sci Help with solving first order ODE using a simple Fortran code, please

    I am trying to solve the following first order ODE using a simple Fortran code : $$ ds/dt=k_i * \sqrt{v}$$ where both (ki) and (v) are variables depending on (h) as follows $$ k_i=\sqrt{χ/h^2}$$ $$v= \mu h$$ where (μ) and (χ) are constants. (the arbitrary values of each of them can be seen...
  45. 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...
  46. 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...
  47. M

    A Rayleigh quotient Eigenvalues for a simple ODE

    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...
  48. 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...
  49. m4r35n357

    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...
  50. W

    Nonlinear first-order ODE?

    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...
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