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.

View More On Wikipedia.org
  1. I

    MHB A property of solution of ODE y''+p(x)y=0

    Let $f$ be a solution of the following equation $y''+p(x)y=0$, $p$ is continuous on $\mathbb{R}$ such that $p(x)\leq 0$ for all $x\in\mathbb{R}$. Suppose that $f$ is defined on $[a,+\infty)$, $f(a)>0$, $f'(a)>0$, $a\in\mathbb{R}$ . Prove $f(x)>0$ for all $x\in[a,\infty)$. Any help would be...
  2. H

    Convert 2nd order ODE to system of 1st order

    Homework Statement Convert the following second-order differential equation into a system of first-order equations and solve y(1) and y'(1) with 4th-order Runge-kutta for h=0.5. ##y''(t)+sin(y(t))=0,\ y(0)=1,\ y'(0)=0## Homework Equations The Runge-kutta method might be applicable, but I know...
  3. I

    Laplace Transform of A Second Order ODE

    23.) y'' + 2y' + y = 4e-t; y(0) = 2, y'(0) = -1 Y(s) = [(as + b) y(0) + a y'(0) + F(s)]/(as2 + bs + c) My attempt: a = 1, b = 2, c = 1 F(s) = 4 L{ e-t } = 4/(s+1) (From Laplace Transform Table) Plugging and simplifying: Y(s) = (2s2 + 5s + 7)/[(s + 1)(s2 + 2s + 1) Here is where I get...
  4. J

    Imposing normalization in numerical solution of of ODE

    Hello all, I would like to know how to impose a normalization condition to numerically solving an ODE. For simplicity let's consider the example \frac{dy}{dx}=y You could use different methods using an initial value, but if you consider the interval [x_0,x_1] and \int_{x_0}^{x_1} y(x)dx=1...
  5. enot

    Parameter for a saddle point

    Hello, this is my first post here, so if I do problems, please correct me and do not be upset = ) I have one small theoretical and one greater question. small one first: Homework Statement I have a potential energy : $$W(L)= -\frac{1}{4}k_4(L_x^4+L_y^4+L_z^4)$$ How can describe my potential...
  6. S

    Substitution to convert first order ODE to homogenous

    Homework Statement Use the substitution ##x=X+h## and ##y=Y+k## to transform the equation ##\frac{dy}{dx}=\frac{2x+y-3}{x-2y+1}## to the homogenous equation ##\frac{dY}{dX}=\frac{2X+Y}{X-2Y}## Find h and k and then solve the given equation Homework EquationsThe Attempt at a Solution If I...
  7. M

    Variational solutions to non-linear ODE

    Hi PF! I have a system of nonlinear ODE's, wherein the only constant ##C## in the ODE takes on several values depending on the geometry; thus once a geometry is defined for the ODE, ##C## is uniquely determined. Let's say I want to guess a quadratic solution to the ODE, call it ##\phi(x)##...
  8. manifold

    A numerical solution of a second order ODE

    Hello everyone; i'd like some help in this problem : i want to solve num this differential equation { y"(t)+t*cos(y)=y } by the Taylor method second order expansion. i first have to make this a first order differential equation by taking this vector Z=[y' y] then we have Z'=[y" y'] which equal...
  9. M

    Are there closed curve solutions for these ODE constraints?

    Are there closed curve solutions for ##\mathbf{v}(t) \in \mathbb{R}^3## satisfying this constraint? $$\mathbf{v}(t) \cdot \frac{d^2}{dt^2}\mathbf{v} = 0 $$
  10. M

    How do I numerically solve a non-linear ODE in Mathematica?

    Hi PF! I am wondering if any of you have experience numerically solving second order ODE's? Basically, I'm trying to solve one and am trying to do it numerically in mathematica. Can anyone help? For those curious, the equation is ##y y'' + 2y'^2 +xy' = 0## where ##y## is a function of ##x##...
  11. LeoDimieri

    Mathematica Solving ODE numerically in Mathematica - I get 'ndnum' error

    Hello everyone! I'm trying hard to solve numerically a system of coupled differential equations of first order, but I get this error everytime.. I can't find the reason.. maybe you can help me, I'd really apreciate that. This is the code: Jg=0.000043; Kg=0.5; Bg=0.06; Bpm=0.5; r=0.11...
  12. I

    MHB How to approach this 2nd order linear ODE?

    $y''+\frac{1}{x}y'=\frac{2}{x^2}-4$ Hey. This is probably really simple but I'm stuck :p how do I approach this?
  13. I

    MATLAB How can I solve a 2nd order ODE in matrix form using Matlab?

    I'm trying to solve a 2nd order differential equation in matrix form. I'm not familiar with Matlab, and have tried solving this using tutorials on youtube. Initially, theta1 = pi/4, theta2 = 7*pi/12, theta1_d = 0, and theta2_d =0. Time interval is (0,1.2). When I try to solve this using ode45...
  14. P

    Power series solution to degree 2 ODE

    Homework Statement (x+1)y'' - (x-1)y' - y = 0 centred around x=1 y(1) = 2, y'(1) = 3 The Attempt at a Solution I know I am supposed to get two power series, one with a0 and one with a1 but when I am trying to figure out a pattern, I keep getting both a0 and a1 in all of my terms. So I end up...
  15. G

    ODE with Dirac Delta and conditions at infinity

    I'm trying to solve the following equation (even if I'm not sure if it's well posed) \partial_{x} \, y(x) + a(x)\, y(x) = \delta(x) with ##\quad \lim_{x \rightarrow \pm \infty}y(x) = 0## It would be a classical first order ODE If it were not for the boundary conditions and the Dirac...
  16. F

    MHB Finding interval where second order ODE has unique solution

    I'm a little stuck getting started on this question. y''+\tan(x)y=e^x with y(0)=1,y'(0)=0. I know the existence and uniqueness theorem for an nth order initial value problem How do I apply the theorem?
  17. M

    Solving Non-linear First Order ODEs with Variable Coefficients?

    Homework Statement $$y' y + \frac{y}{x} = 1 - 2x$$ Homework Equations nothing comes to mind The Attempt at a Solution i've guessed a quadratic but that didn't work. now I'm stuck. any ideas? also, this is not homework, but a problem I am working on. Thanks!
  18. J

    Solve the ODE with initial condition:

    y''-10y'+25=0 Solve the ODE with initial condition: y(0) = 0, y' (1) = 12e^5 . I keep getting y=12/5e^5x when c1=0 and c2=12/5 ... but Answer key says y=2xe^5x what am I doing wrong?
  19. C

    Linearity in differential equations

    Homework Statement Is the equation (x2sinx + 4y) dx + x dy=0 linear This problem also asks me to solve it, but I don't have a problem with that part. Homework Equations An equation is linear if the function or its derivative are only raised to the first power and not multiplied by each other...
  20. evinda

    MHB Why ODE Linear? Initial Value Problem Explanation

    Hello! (Wave) I am looking at initial value problems for ordinary differential equations.Let $a,b, \ a<b, \ f: [a,b] \times \mathbb{R} \to \mathbb{R}$ function and $y_0 \in \mathbb{R}$.We are looking for a $y: [a,b] \to \mathbb{R}$ such that$$(1)\left\{\begin{matrix} y'(t)=f(t,y(t))\\ y(a)=y_0...
  21. L

    Nonlinear Ordinary Differential Equation Help

    Homework Statement y'=(x^2 +xy-y)/((x^2(y)) -2x^2)[/B]Homework EquationsThe Attempt at a Solution I know that really the only way to solve this one is to use an integrating factor, and make it into an exact equation. My DE teacher said that to make it into a exact equation you need to take...
  22. A

    Converge pointwise with full Fourier series

    I am working on a simple PDE problem on full Fourier series like this: Given this piecewise function, ##f(x) = \begin{cases} e^x, &-1 \leq x \leq 0 \\ mx + b, &0 \leq x \leq 1.\\ \end{cases}## Without computing any Fourier coefficients, find any values of ##m## and ##b##, if there is any...
  23. K

    Checking if an equation is exact and finding the solution

    Homework Statement Use the "mixed partials" check to see if the following differential equation is exact. If it is exact find a function F(x,y) whose differential, dF(xy) is the left hand side of the differential equation. That is, level curves F(xy)=C are solutions to the differential...
  24. F

    Determining the tension on a rotating particle

    Homework Statement [/B] A particle of mass m slides (both sideways and radially) on a smooth frictionless horizontal table. It is attached to a cord that is being pulled downwards at a prescribed constant speed v by a force T (T may be varying) Use F=ma in polar coordinates to derive an...
  25. M

    Solving a Homework Equation: (3xy^2+4y)dx+(3x^2y+4x)dy=0

    Homework Statement (3xy^2+4y)dx+(3x^2y+4x)dy=0Homework Equations The Attempt at a Solution So First I checked if both equations were exact. I took the derivative of 3xy^2+4y and also derivative of the other and they were both equal so the equation is exact. I took the 3xy^2+4y and integrated...
  26. P

    First order ODE, orthogonal trajectories

    1. The problem statement, all variables and given/known da ##\frac{x^{2}}{k^{2}} + \frac{y^{2}}{\frac{k^{4}}{4}} = 1## with k != 0 this can be simplified to ##x^{2} + 4y^{2} = k^{2}## Find dy/dx implicitly, then find the new dy/dx if you want orthogonal trajectories to the ellipse. Lastly solve...
  27. M

    MATLAB Matlab finite difference schemes

    I have big problem with finite difference schemes (DS) on Matlab. I need write DS on Matlab, example: u_x=(u_(i+1,j)-u_(i-1,j))/2, we choose step is 1. On Matlab: u_x=(u( :,[2:n,n])-u( :,[1,1:n-1]))/2 And I can write u_y, u_xx, u_yy, u_xy. But now, I need to write for higher order, example...
  28. S

    General solution of second order ODE

    Homework Statement Find the general solution. Homework Equations y"+y=x2sin2x The Attempt at a Solution Characteristic equation would be: m2 + 1 = 0 So,m2 = -1 Therefore, m = i or m = -i. Complementary function would be : Asinx+Bcosx where,A and B are constants respectively. If I write...
  29. L

    Encountering "unsolvable equation" during simple ODE problem

    Homework Statement {Prob #27, Section 2.1 "EDE" (Boyce/Prima, 10thEd), pp. 40} "Consider the initial value problem [and] find the coordinates for the first local maximum point of the solution, t>0." Homework Equations y' + (1/2)y = 2cost y(0) = -1 The Attempt at a Solution I...
  30. G

    Solve ODE: (x^2+2)y'''-(x^2+2x+2)y''+2(x+1)y'-2y=0

    Hello again, I've got another trouble with a new differential equation: (x^2+2)y'''-(x^2+2x+2)y''+2(x+1)y'-2y=0 I did a try using matrices(file is attached), however the system to be solved is hard to compute. Do you have any idea to help me solve this out?
  31. S

    Solving an ODE using shooting method

    Hi, I am trying to solve the following ODE for my maths project: ## y'' = \frac{\alpha}{2}y^3 - \frac{3}{2}y^2 + y - \frac{3}{x} y'## under the following boundary conditions: ## y'(0) = 0 ## ## y(x) \rightarrow y \_ \equiv 0\ \text{as}\ x \rightarrow \infty ## As a first step, I converted...
  32. C

    General Solution to a certain form of ODE

    While fiddling around with some very simple linear ODEs, I "discovered" a formula that gives a solution to ODEs of the form: ##y'+y=ax^n ##. here it is: i'm sure that this was discovered before, but i was just wondering if it had any official name or something.
  33. M

    ODE Uniqueness: Exploring the Solution of Differential Equations

    Is the solution of differential equation be unique always?
  34. K

    Why does the 2nd order homogeneous linear ODE have 2 general solutions?

    why not the 2nd order linear homogeneous ODEs have three Linearly independent solutions or more? I know for the characteristic equation, we can only find 2 answers but.. just wondering if that is the only case to solve the question and if it is, then why it has to be. so my question is,1. 2nd...
  35. C

    Solving an ODE with variable coefficients

    Homework Statement Solve the following: [/B] y'' = c2 / (x2 + c1*x) * y c1, c2 are constants, x is variableHomework Equations As above The Attempt at a Solution I have used the method of Frobenius and regular power series and obtained an infinite series on top of an infinite series, which is...
  36. topsquark

    MHB Approximate solution for non-linear ODE

    I've decided to finish off this stage of my GR problem by finding an interval over which the acceleration of the object is "roughly" constant. I don't need help with the Math per se, but I would like your opinion on the method I am proposing. The Math is sufficiently ugly that I'd like some...
  37. T

    Am I rewriting this differential correctly?

    Homework Statement I have a differential equation that I need to solve numerically by writing a program. x0, y0, x_dot0, y_dot0, α are all given Hello, I have the following differential equation: http://puu.sh/d78KC/107bd6c71f.png I want to rewrite it so I can solve it numerically by writing...
  38. icesalmon

    Second order ODE for RLC circuit

    if I consider a circuit consisting of a capacitor, an inductor and a resistor and using kirchhoffs voltage rule for the circuit i come up with the following L(Q''(t)) + R(Q'(t)) + (Q(t))/C = 0 I solve for the roots using a characteristic equation of the form LM2 +MR +(1/C) = 0 solving this for...
  39. V

    Numerical Solution for ODE with Critical Point: A General Procedure Guide

    Hello, can someone advise me how to solve numerically ODE which consist of function with "critical point" (Im not sure if it is good definition)? I mean for example this one: y'(x)=\frac{\sin{x}}{x}, where in x=0 has function a "problem". I know that limit ->1 but in numerical solutions it...
  40. C

    How Can We Solve this Second Order ODE for Electron Behavior?

    Homework Statement I'm taking an online introductory chem course, and while explaing the failure of classical mechanics to describe electron behavior, the teacher brought up the following ode which is based on Newton's second law and coulombs law: -e^2/4(pi)(epsilon-nuaght)r^2=m(d^2r/dt^2)...
  41. C

    First Order ODE With v = y' Substitution

    Homework Statement 8t^2 * y'' + (y')^3 = 8ty' , t > 0 Homework EquationsThe Attempt at a Solution I tried using the substitution v = y' to get: 8t^2 * v' + v^3 = 8tv I rewrote it in the form 8t^2 * dv/dt + v^3 = 8tv, and then moved the v^3 to the other side to get 8t^2 * dv/dt = 8tv - v^3...
  42. F

    MHB Linear ode; how does answer key get this?

    Solve the linear equation: x\frac{dy}{dx}-y=x^2sinx rewrite \frac{dy}{dx}-\frac{y}{x}=\frac{x^2sinx}{x}=xsinx P(x)=\frac{-1}{x} So e^ { \int \frac{-1}{x} dx }=-1<=this is where I went wrong \frac{d}{dx}[-y]=-xsinx \int -x sin(x)=xcosx-sinx +C but the answer key gives y=cx-xcosx
  43. D

    Differential Equations: Bernoulli Equation

    Homework Statement Find the general solution: y'-3y=(y^2) Homework EquationsThe Attempt at a Solution divide both sides by y^2 y'(y^-2) -3(y^-1) = 1 we know v=y^(n-1) v=y^-1 v'=d/dx(y^-1) v'=-(y^-2) y' plug it back into y'(y^-2) -3(y^-1) = 1 -v'-3v=1 this is where I think I am making a...
  44. R

    Help: Analytical Solution to Coupled ODE

    Hello , I tried to solve this coupled ODE but with no success Does anyone know if there is an analytical solution to this equation? my problem is with the first & the second equations the term g*f is the my biggest problem i think once i have the solution for g - the solution for h is...
  45. D

    ODE with base excitation caused by a half sine wave

    Homework Statement The suspension system of a car traveling on a bumpy road has a stiffness of ##k = 5\times 10^6## N/m and the effective mass of the car on the suspension is ##m = 750## kg. The road bumps can be considered to be periodic half sine waves with period ##\tau##. Determine the...
  46. C

    Oddly Formatted Second Order ODE

    Homework Statement u'' + w20*u = cos(wt) w refers to omega. Homework EquationsThe Attempt at a Solution I'm not sure where to begin on this. For starters, it's a multiple choice problem, and all the answers are given in terms of y, so I'm not sure if u is supposed to replace y' or something...
  47. _N3WTON_

    Second Order ODE, Complex Roots, Change of Variables

    Homework Statement Solve: \frac{d^{2}y}{dx^{2}} + \omega^{2}y = 0 Show that the general solution can be written in the form: y(x) = A\sin(\omega x + \alpha) Where A and alpha are arbitrary constants Homework EquationsThe Attempt at a Solution I know that I will need to change variables for...
  48. C

    Second Order ODE, With Initial Conditions

    Homework Statement y'' + 4y = t2 + 6et; y(0) = 0; y'(0) = 5 Homework Equations The Attempt at a Solution So, getting the general solution, we have r2 + 4 = 0, so r = +/- 2i So the general solution is yc = sin(2t) + cos(2t) I then used the method of undetermined coefficients to figure that...
  49. T

    Finding the Fundamental Solutions of a Third Order ODE

    Homework Statement Find the fundamental solution to ty''' - y'' = 0 Homework EquationsThe Attempt at a Solution I think I'm missing something really obvious, but I have the characteristic polynomial: t\lambda^3 - \lambda^2 = 0 Solving the equation: \lambda^2 (t\lambda - 1) = 0 I get zero...
  50. F

    MHB How to solve a differential equation using substitution?

    Solve the DE using an appropriate substitution. (x-y)dx+xdy=0 First step is to determine the substitution. I was told for homogeneous ODEs to always make the substitution y=ux but the substitution u=x-y looks better. Let u = x-y then u'=-y' which means y'=-u' rewrite the original equation...
Back
Top