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. gurbir_s

    I Rosenbrock method ODE : Abrupt increase during First iteration

    The set of equations is I have first tried to solve only first two equations (removing the components of other 4 equations from them.) This is the output, where first column is the time, 2nd - X_p, and 3rd - X_n
  2. F

    I ODE with non-exact solution: closed-form, non-iterative approximations

    In case of an integral ##\rightarrow## differential equation of the type: $$ f(t) = \int_0^t g(f(\tau)) d\tau $$ $$ \rightarrow \frac{df(t)}{dt} = g(f(t)) $$ which turns out not to be solvable in exact form because ##g(f(t))## is a non-polynomial function (but it would if ##g(f(t))## was a...
  3. B

    A Seeking advice about solving an ODE

    Hello! I am trying to solve the time dependent Schrodinger equation for a 2x2 system and I ended up with this ODE: $$y''=-iA\sin{(\omega t)}y'-B^2y$$ with the initial conditions ##y(t=0)=0## and ##y'(t=0)=B##. I can look at it numerically but I was wondering if there is a way to get something...
  4. Euge

    POTW Solve Linear System of ODEs: x',y' w/ Initial Conditions x(0),y(0)

    Solve the linear system of ODE ##x' = 2x + 3y##, ##y' = -3x + y## with initial conditions ##x(0) = 1, y(0) = 2##.
  5. ka_reem13

    Stuck on a 1st order ODE

    I'm aware that I can introduce the perimeter p = dy/dx then I can rearrange my equation to make y the subject, then I can show that dp/dx = p/x. However, this only gives me a bunch of quadratic curves for my solution. However given part b I see that two curves are meant to intersect each point...
  6. V

    Solving An IVP on Matlab with ODE 45 with different tolerances

    My code is as follows: but when I use the function in my command window exactsol(t) and input a tolerance but there is an error in LINE 19 saying unrecognized ivpfun, could someone help me fix it as I am unsure of how to proceed from here. function y = exactsol(t) y = zeros (2,1); y(1) =...
  7. PhysicsRock

    Leaking cart being accelerated

    My approach is to use the definition of the Force with ##\displaystyle F = \frac{dp}{dt} = \dot{m} v + m \dot{v}##. Since ##m(t)## decreases linearly, I should be able to set ##m(t) = M - \Phi t##, thus ##F = - \Phi v + (M - \Phi t) \dot{v}##, which gives ##\displaystyle v = -\frac{ F - (M -...
  8. H

    Mr. Tenenbaum's and Prof. Mattuck's advice not working (ODE)

    All right, we got $$ y'' + y = 4x \sin x $$ We are doing the Complexification $$ \tilde{y''} + \tilde{y} = 4x e^{ix} $$ Complementary function: $$ \begin{align*} \textrm{characteristic equation =}\\ m^2 + 1 = 0 \\ m = \pm i \\ \tilde{y_c} = c_1 e^{ix} + c_2 e^{-ix} \\ \end{align*} $$ Q(x)...
  9. H

    Repeated roots of a characteristic equation of third order ODE

    The characteristic equation ## m^3 -6m^2 + 12m -8 = 0## has just one single, I mean all three are equal, root ##m=2##. So, one of the particular solution is ##y_1 = e^{2x}##. How can we find the other two? The technique ##y_2 = u(x) e^{2x}## doesn't seem to work, and even if it were to work how...
  10. K

    Conservation of probability issue when solving ODE in Mathematica

    I am trying to solve this two level (Schrodinger) equation as a function of time: $$i\begin{pmatrix} \dot{x}\\ \dot{y} \end{pmatrix} = \begin{pmatrix} 0 & iW+dE_0sin(\omega t)\\ -iW+dE_0sin(\omega t) & \Delta \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}$$ (I can go into more details about...
  11. T

    I Second Order ODE with Exponential Coefficients

    Hi all, I have another second order ODE that I need help with simplifying/solving: ##p''(x) - D\frac{e^{\gamma x}}{A-Ae^{\gamma x}}p'(x) - Fp(x) = 0## where ##\gamma,A,F## can all be assumed to be nonzero real numbers and ##D## is a purely nonzero imaginary number. Any help would be appreciated!
  12. F

    I Real ODE yields real solution through complex numbers

    Hello, I'm posting here since what follows is not about homework, but constitutes a personal research which underlies some more general questions. As with the infamous "casus irreducibilis" (i.e. finding the real roots of a cubic function sometimes requires intermediate calculations with...
  13. TGV320

    Courses Math Analysis & ODE Intro: MIT ODE Course?

    Hello, I am currently planning on self studying math analysis with MIT ocw courses, but I cannot find the analysis course. I have found on the web that analysis and ODE are the same thing. If I want to get a good introduction to analysis, is the MIT ODE course fit for such a use? Thanks
  14. J

    Solve this ODE in terms of a power series

    I believe I am doing everything right up until the point where I have to try and find a recurrence relation. I honestly have no idea what to do from there. I've listed my work in getting the powers of n and the indicies to all match. Any help appreciated. Here is the original DE...
  15. T

    I Converting Second Order ODE to Hypergeometric Function

    I believe it is the case that any linear second order ode with at most 3 regular singular points can be transformed into a hypergeometric function. I am trying to solve the following equation for a(x): where E, m, v, k_{y} are all constants and I believe turning it into hypergeometric form will...
  16. T

    Coupled First Order ODE

    Hey all, I am currently struggling decoupling (or just solving) a system of coupled ODEs. The general form I wish to solve is: a'(x)=f(x)a(x)+i*g(x)b(x) b'(x)=i*h(x)a(x)+j(x)a(x) where the ' indicates a derivative with respect to x, i is just the imaginary i, and f(x), g(x), h(x), and j(x) are...
  17. M

    A Green's function for Sturm-Louiville ODE

    Hi PF! Given the following ODE $$(p(x)y')' + q(x)y = 0$$ where ##p(x) = 1-x^2## and ##q(x) = 2-1/(1-x^2)## subject to $$y'(a) + \sec(a)\tan(a)y(a) = 0$$ and $$|y(b)| < \infty,$$ where ##a = \sqrt{1-\cos^2\alpha} : \alpha \in (0,\pi)## and ##b = 1##, what is the Green's function? This is the...
  18. A

    Partial fraction decomposition with Laplace transformation in ODE

    Hello! Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For...
  19. O

    Show that ODE is homogeneous, but I don't think it is

    Ignoring the second part of the question for now, since I think it will be more clear once I understand how this equation is homogeneous. According to my textbook and online resources a first-order ODE is homogeneous when it can be written like so: $$M(x,y) dx + N(x,y) dy = 0$$ and ##M(x,y)##...
  20. M

    A Help with understanding why limit implies uniqueness

    I'm studying ODEs and have understood most of the results of the first chapter of my ODE book, this is still bothers me. Suppose $$\begin{cases} f \in \mathcal{C}(\mathbb{R}) \\ \dot{x} = f(x) \\ x(0) = 0 \\ f(0) = 0 \\ \end{cases}. $$ Then, $$ \lim_{\varepsilon \searrow...
  21. P

    Engineering How do I use Simulink to create a control system with a 2nd order ODE?

    Equation: , where matrix D, C, G and F can be represented by I'm supposed to design a control system that looks like this: I am given that the dynamic model = fcn(D,C,G,dq) where the dq is the same as 𝑞̇ and d2q in the diagram is the same 𝑞̈. The default initial value of [𝑞(0), 𝑞̇(0)] is...
  22. wrobel

    A Lagrange-D’Alembert Principle and random ODE

    Here is my paper. A criticism and other comments are welcome. Abstract: The Lagrange-D'Alembert Principle is one of the fundamental tools of classical mechanics. We generalize this principle to mechanics-like ODE in Banach spaces. As an application we discuss geodesics in infinite dimensional...
  23. T

    A Existence of unique solutions to a first order ODE on this interval

    I am trying to find a way to prove that a certain first order ode has a unique solution on the interval (1,infinity). Usually the way to do this is to show that if x' = f(t,x) (derivative with respect to t), then f(t,x) and the partial derivative with respect to f are continuous. However, this...
  24. chwala

    Solve the homogeneous ODE: dy/dx = (x^2 + y^2)/xy

    this is pretty easy for me to solve, no doubt on that. My question is on the constant. Alternatively, is it correct to have, ##ln x= \frac {v^2}{2}##+ C, then work it from there... secondly, we are 'making" ##c= ln k##, is it for convenience purposes?, supposing i left the constant as it is...
  25. I

    ODE solver for second Order ODE with Stiffness and Mass Matrices

    i am new to MATLAB and and as shown below I have a second order differential equation M*u''+K*u=F(t) where M is the mass matrix and K is the stifness matrix and u is the displacement. and i have to write a code for MATLAB using ODE45 to get a solution for u. there was not so much information on...
  26. M

    MHB Solving an ODE: Seeking c so y'(0)=0

    so I am trying to solve this equation y'+5.6y=9.5cos(2x)+2.4sin(2x) . I want the c in order to y'(0)=0. I am really lost
  27. R

    I Series Solution for 2nd-Order Homogeneous ODE

  28. S

    MATLAB How can I plot a Hohmann Transfer Orbit in MATLAB using ode45?

    function Asteroid_Mining clc %Initial conditions g0 = 9.81; %gravity (m/s^2) p = 1.225; %atmospheric density at sea level (kg/m3) Re = 6378; %radius of Earth (km) Ra = 7.431e7; %distance of Bennu from Earth in (km) [August 2023] G = 6.674e-11/1e9; % Gravitational constant (km3/kg.s2) mu =...
  29. S

    MATLAB Calculating the time taken for an asteroid mission using MATLAB

    Using MATLAB, I am trying to calculate the time taken for a spacecraft to travel from Earth to a near Earth asteroid and then returning back to Earth but so far I have had no luck. Furthermore, I want to plot a Hohmann transfer and calculate the mass of fuel required for this mission. If...
  30. Falgun

    Applied Ordinary Differential Equations Books

    I am trying to self study Ordinary Differential Equations and am totally fed up of "cookbook style ODEs". I have recently finished Hubbard's Multivariable Calculus Book and Strang's Linear algebra book. I would like a rigorous and Comprehensive book on ODEs. I have shortlisted a few books below...
  31. R

    Find a continuous solution to an ODE that includes a step function

    Non-homegenous first order ODE so start with an integrating factor ##\mu## $$\mu=\textrm{exp}\left(\int a dt\right)=e^t.$$ Then rewrite the original equation as $$\frac{d}{dt}\mu y = \mu g(t).$$ Using definite integrals and splitting the integration across the two cases, $$\begin{align}...
  32. docnet

    Converting a PDE to an ODE

    Sorry the problem is a bit long to read. thank you to anyone who comments. We consider the initial value problem for the Burger's equation with viscosity given by $$\begin{cases} \partial_t u-\partial^2_xu+u\partial_xu=0 & \text{in}\quad (1,T)\times R\\\quad \quad \quad \quad \quad...
  33. Mathman2013

    Lower bound of an ODE?

    I have following differential equation dV/dt = 5 - 2 * V(t)^(1/3) which represents a the time its take to drain a barrel of rain water which contain 25 Liter of water, at t = 0. I am suppose to calculate the least amount of water in barrel during this process. If I set the rate of growth to...
  34. V

    Using an ODE to show a local minimum

    The ODE given to us is y' = xcosy. I am having a bit of trouble when it comes to solving this problem. We are supposed to show that the solution has a local minimum at x = 0 with the hint to think of the first derivative test. However, I am only really familiar with the first derivative test...
  35. V

    Find two solutions for an ODE

    I have an ODE, which is y' = 5x3(y-1)1/5 with the initial condition y(0)=1, I must find two solutions. My attempt at solving this problem is as follows: Separate the equation, we get, dy/(y-1)1/5 = 5x3dx. Integrate both sides, ∫ dy/(y-1)1/5 = ∫ 5x3dx. We are left with, (5/4)(y-1)4/5 = (5x4)/4...
  36. V

    Existence and Uniqueness For ODE

    I'm new to learning about ODE's and I just want to make sure I am on the right track and understanding everything properly. We have our ODE which is y' = 6x3(y-1)1/6 with y(x0)=y0. I know that existence means that if f is continuous on an open rectangle that contains (x0, y0) then the IVP has...
  37. Tony Hau

    How to solve this 2nd order ODE?

    This is a very simple question: I would like to solve for ##\psi## in this equation $$\frac{d^{2}\psi}{d\xi^2} =\xi^2\psi$$ I so apply ##y=c_{1}e^{-kx}+c_{2}e^{kx}## and ##\psi## should be equal to ##\psi=c_{1}e^{-\xi^2}+c_{2}e^{\xi^2}##, because ##(D^2-\xi^2)\psi=0##. However the answer is...
  38. Leonardo Machado

    A Improper boundary in non-linear ODE (pseudospectral methods)

    Hello, I am trying to compute some non-linear equations with pseudospectral/collocation methods. Basically I am expanding the solution as $$ y(x)=\sum_{n=0}^{N-1} a_n T_n(x), $$ Being the basis an Chebyshev polynomial with the mapping x in [0,inf]. Then we put this into a general...
  39. A

    I Non-linear ODE: initial conditions

    Say you have the set of coupled, non-linear ODEs as derived in this thread, it has two unknowns ##N(t)## and ##\theta(t)##: $$ N - mg = - m\frac{L}{2}\left(\dot{\theta}^2\cos(\theta) + \ddot{\theta}\sin(\theta)\right)$$ $$ \frac{L}{2}N\sin(\theta) = \frac{1}{12}ml^2\ddot{\theta}$$ What freedom...
  40. K

    I Help - Derivation of Pulsating Star Euler ODE

    to I am a bit clueless on how to get break the ##r X(r)## from inside the derivative. P.S. I tried to copy from Symbolab instead of pasting the picture, but it didn't let me.
  41. M

    A Solution to an ODE (Leaky Integrate and Fire Neuron Model)

    I'm looking for a general analytical solution to a particular ODE that comes up in neuroscience a lot. My feeling is that such a solution can't be obtained, otherwise someone would have presented it by now, but I don't have a good understanding of why it is so hard to solve. The equations as...
  42. O

    I Derive local truncation error for the Improved Euler Method

    I'm trying to find the local truncation error of the autonomous ODE: fx/ft = f(x). I know that the error is |x(t1) − x1|, but I can't successfully figure out the Taylor expansion to get to the answer, which I believe is O(h^3). Any help would be greatly appreciated!
  43. Kaguro

    Comp Sci Numerically, how to get the other solution of this 2nd order ODE?

    Actually I was trying to write a small program in Scilab to simulate a quantum particle. When I give a potential higher than energy, the wave function should go like exp(-x) and decay. But my program just increases without bound. Is there any nice way to do anything about it?
  44. jisbon

    Engineering Heun's Method for Second Order ODE (Eng Maths)

    Question: So I got around on doing this example, and I'm pretty sure I messed up somewhere, would appreciate if someone could point out what I did wrongly. 1) For any second ODE, I should let: ##y_{1}= y ## and ##y_{2}= y' ## Hence, ##y_{1}'= y' = y_{2} ## and ##y_{2}'= y'' = xy(x)+x^2-y(x) =...
  45. agnimusayoti

    Another Second Order ODE Problem (ML Boas, Ch 8, Sec 7, Prob 5)

    With the new variable, I got: $$p^2 (p'_y)^{2}=k^2(1+p^2)$$ where ##p'_y## is ##\frac{dp}{dy}##. I modified the equation so the variable p and dp can be separated from dy. Here what I got: $$\frac{p}{\sqrt{p^2+1}} dp=k dy$$ I substitute ##p^2+1=u## so I got $$\sqrt{u}=ky+c_1$$ Back substitution...
  46. Kaguro

    Find the general solution of this nonlinear ODE: xy' + sin(2y) = x^3*sin^2(y)

    This equation, is non linear, non-separable, and weird. I would like to have a direction to start working on this. I tried writing sin(2y) = 2sin(y)*cos(y). See, ##xy' = x^3sin^2(y)-2sin(y)cos(y)## Can't separate. Writing in this way: ##(x^3sin^2y-sin2y)dx-xdy=0## Also, I checked that it is...
  47. S

    Laplace transform of an ODE with a non-smooth forcing function

    Suppose I'm solving $$y''(t) = x''(t)$$ where $$x(t)$$ is the ramp function. Then, by taking the Laplace transform of both sides, I need to know $x'(0)$ which is discontinuous. What is the appropriate technique to use here?
  48. 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...
  49. 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...