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. 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...
  2. 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...
  3. 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...
  4. J

    Rearrange equation (solution of ODE)

    I have determined the solution to a nonlinear first order ordinary differential equation but am struggling to rearrange the result, I have that $$\\ln(R)+\frac{mR^{n-1}}{n-1}=\bar{w}_{\infty}\xi+C.$$ How would I rearrange this equation for $$R$$?
  5. _N3WTON_

    Second-Order ODE with Missing X

    Homework Statement Solve the given differential equations with missing x. y'' + y = 0 Homework Equations y = c_1cos(x) + c_2sin(x) This is the answer given in the back of the book. However, I can't sem to get my answer to agree The Attempt at a Solution First, I made some substitutions: y'...
  6. C

    Analytically solve second-order coupled ODE (damping term coupled)

    Homework Statement I need to (analytically) solve a system of coupled second-order ODEs: (A) \frac{du}{dt} - fv = \Omega^2x (B) \frac{dv}{dt} + fu = \Omega^2y where u = \frac{dx}{dt} v = \frac{dy}{dt} subject to the initial conditions u(t=0) = U and v(t=0) = 0. Homework Equations --- The...
  7. 2

    Second order nonlinear ODE. How to begin solving it?

    Homework Statement This is not the exact problem that I want to solve but I will use this as a guidance tool: ##y'' - (y')^2 + y^3 = 0## where y is the function of x 2. The attempt at a solution I tried doing a substitution ##u(x) = y'(x)## which leads to ##u' - u^2 + y^3 = 0## where both u...
  8. R

    Solving a second-order linear ODE in an infinite limit

    Homework Statement So this is part of a broader problem about the quantum harmonic oscillator, but there's one particular bit of mathematics I'm stuck on. We have the differential equation: y''(x) +(ε-x2) y = 0 And I'm told that we're to examine how y behaves as x tends towards...
  9. G

    How can I simplify these ODEs?

    Homework Statement the first one y'=\frac{y^{2}+xy^{2}}{x^{2}y-x^{2}} the second one xyy'=\frac{x^{2}+1}{y+1} Homework Equations The Attempt at a Solution i separated x and y variable then integrate both of them in the first one ∫\frac{y-1}{y^{2}}dy=∫\frac{1+x}{x^{2}}dx...
  10. F

    MHB Solve 1st-Order ODE: Chicken in 375° Oven

    I'm going to need a little help with this one. I get an answer but it doesn't make sense. The question states According to Newton's law of cooling, the time rate of change of temperature T(t) of a body immersed in a medium of constant temperature A is proportional to the difference A-T. That...
  11. F

    2nd Order Nonlinear ODE Question

    Homework Statement Solve the 2nd order nonlinear differential equation, with initial conditions y(0)=0 and y'(0)=1 y''=2ay^3-(a+1)y with a within [0,1] It would be greatly appreciated if someone could point me in the right direction on this. Thanks! Homework Equations The Attempt at a...
  12. D

    MHB How Can I Solve This Second Order Linear ODE Problem?

    I'm having a lot of trouble with this problem. I'm also having a lot of trouble inputting it into LaTeX. I hope you can follow even though the markup isn't good. I'm trying to find a formula for the general solution of $ax^2y''+bxy'+cy=0$ where $y=x^r\ln(x)$ when $(b-a)^2-4ac=0$; using...
  13. N

    MHB How Do You Calculate This Indefinite Integral in a First Order Linear ODE?

    Hi How do you calculate the following indefinite integral: \int \frac{2x}{125+3t} dt a step by step solution would be appreciated
  14. D

    Derivation step in ODE solution in textbook

    This is not homework but is part of the solution process of an ODE and I cannot understand how the author made a derivation step. After a change of variable in the original ODE, the ODE in the new independent variable has a standard method of solution. But instead of using this method, the...
  15. J

    What is the Definition of a Linear ODE?

    I am learning how to solve 1st order linear ODEs using the integrating factor.However, I run into confusion at the definition of a linear ODE. According to a reliable source, a linear ODE must have the form: (dy/dt) + p(t)y= g(t) I don't understand what it means for an ODE to be...
  16. D

    Solving this ODE for an initial value problem

    Homework Statement x \frac{du}{dx} \ = \ (u-x)^3 + u solve for u(x) and use u(1) \ = \ 10 to solve for u without a constant. Homework Equations The given hint is to let v=u-x The Attempt at a Solution This equation is not separable and the book wants me to make it separable...
  17. gfd43tg

    Solving higher order ODE as system of first order

    For this problem, I am stuck on the actual system. I don't see what substitution I can make, and the fact that ##u(v)## is a piece-wise function is tripping me up. How the heck do I approach this?? This doesn't look like a standard problem at all.
  18. mazgan

    Set of 2 nonlinear ODE in mathamtica 9

    i just signed up here so i hope this is the right place. i need to solve a set of 2 non-linear ordinary differencial equations. i tryed using NDSolve but it doesn't really work so I am not sure what's wrong with my code. here is my code (copy paste): c = 0.1; Subscript[sys, B]...
  19. M

    Solve 1st Order ODE Homework - Get Help Now

    Homework Statement Solve the below differential equation Homework Equations The Attempt at a Solution I have attached my attempt at solution. But I don't how to get rid of (ln y) term in my equation i.e, i Don't know how to write in terms of y. Please help
  20. W

    Solving a First Order Inhomogeneous ODE for a Regular LR Circuit with AC Voltage

    For a regular LR circuit (L and R in series) and with a AC voltage: I tried to derive the solution myself. https://www.dropbox.com/s/jmsu9j0vt91ze8x/LRcircuit.jpg So first I solved with undetermined coefficients, plugged them in, and then solved with Cramer's Rule. Then I added...
  21. J

    Can an ODE accurately model an RLC circuit with additional sources?

    The equation for this physical model is: http://upload.wikimedia.org/wikipedia/commons/f/fb/RLC_series_circuit_v1.svg And for this is: http://upload.wikimedia.org/wikipedia/commons/d/d0/RLC_parallel_circuit_v1.svg But and if now I add a source of current in those schemes, the...
  22. M

    How to Solve a System of Non-Linear ODEs in Physics?

    Hello everybody. Solving a problem in Physics I run into a system of equations that I do not know how to solve, I would appreciate some help. Here is the system: \ddot{x}+4\dot{x}^2=C_1e^{y} \dot{y}^2=C_2\ddot{x} The dependent variables are x,y. C_1 and C_2 are some constants. I try...
  23. J

    Are General Solutions of Linear ODEs Always Equivalent?

    Given the following ODE: ##ay''(t) + by'(t) + cy(t) = 0## The following solution: ##y(t) = c_1 \exp(x_1 t) + c_2 \exp(x_2 t)## is more general than: ##y(t) = A \exp(\sigma t) \cos(\omega t - \varphi)## ? Why?
  24. J

    Problem with finding second solution to ODE

    1. The problem is to find the series solution to the following differential equation $$ x^2 \frac{d^2 x}{dx^2}+x\frac{dy}{dx}+(x^2 - 1)y $$ 3. Using the ansatz $$ y = \sum _{\lambda = 0}^{\infty}a_{\lambda}x^{k+\lambda}$$ the solution to the indicial equation was found to be...
  25. O

    ODE textbook recommentation (Arnold or other?)

    Hi everyone. I have a copy of Ordinary Differential Equations by Vladimir Arnold. I'm hoping to learn more about differential equations, building up to differential equations on manifolds. I've heard that this is a great book, but I've also heard Arnold sometimes leaves out important details...
  26. M

    ODE Problem: Solution for y'' + (1/x)y' - lambda y = 0 with Boundary Conditions

    Homework Statement $$y'' + \frac{1}{x}y' - \lambda y = 0$$ where ##x \to \infty \implies y \to 0## and ##x \to 0 \implies y' \to 0##The Attempt at a Solution to begin, this was initially a pde, and I've applied separation of variables. to solve this ODE, it seems i cannot assume ##y=e^{rx}##...
  27. J

    Can a 2nd Order Linear ODE Be Expressed in Terms of σ and ω?

    If a 2nd order linear ODE: can written in terms of natural frequency ω0 and damping ratio ζ: being: So, it too can be written in terms of exponential decay/growth σ and angular frequency ω?
  28. N

    Attenuation numerical instability via numerical diss. nonlinear ODE

    Hi, I'm solving a problem numerically that takes the form Q_{ij} \ddot{y}_j +S_{ijk}\dot{y}_j\dot{y}_k +V_i=0, where (Q_{ij},S_{ijk},V_i) are all functions of the dependent variables y_i. The dependent variables are all functions of the variable t. The resolution of this spectral...
  29. C

    2nd order ODE - undetermined coefficients?

    Homework Statement 1) Find the general solution of y''+ω02=Ccos3(ωx) 2) Show there exists two frequencies at which resonance occurs and determine them The Attempt at a Solution I've tried the method of undetermined coefficients, assuming a solution of the form y=(Acos(ωx)+Bsin(ωx))3...
  30. O

    What is the solution to this ODE (and SDE)?

    I'm trying to analyze the following Ito stochastic differential equation: $$dX_t = \|X_t\|dW_t$$ where X_t, dX_t, W_t, dW_t \in \mathbb{R}^n. Here, dW_t is the standard Wiener process and \|\bullet\| is the L^2 norm. I'm not sure if this has an analytical solution, but I am hoping to at...
  31. M

    Solving an ODE: Seeking Suggestions

    Hello I am trying to solve this ODE dx/dt=(f(x)+g(t))^(1/2) I have been recalling what I learn in my ODE course and looking at my old textbook but I did not find what method is appropiate to try...any suggestions? Thank you very much!
  32. MarkFL

    MHB Jaganmaya's question at Yahoo Answers regarding a first order ODE

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  33. T

    You can edit the title of the first post and add [solved] at the beginning.

    I'm confused by problem 2.31 in mathematical tools for physics. Problem: 2.31 The Doppler effect for sound with a moving source and for a moving observer have different formulas. The Doppler effect for light, including relativistic effects is different still. Show that for low speeds they are...
  34. MarkFL

    MHB Solve Linear Inhomogeneous 2nd Order ODE - Alvin's Question on Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  35. J

    What is the difference between these types of differential equations?

    Which the difference between diff equations of kind: \frac{dy}{dx} = \exp(x) \frac{dy}{dx} = 1/x and diff equations of kind: \frac{dy}{dx} = y \frac{dy}{dx} = \frac{1}{\exp(y)} ?
  36. J

    Why a explicit ODE is explicited with y of higher grade?

    Given a implicit ODE like F(x, y(x), y'(x), y''(x)) = 0, why your explicit form is y''(x) = f(x, y(x), y'(x))? Why a ODE is explicited always with y of higher grade?
  37. quasar987

    2nd order homogenous constant coefficient ODE question

    If we have a constant coefficient second order homogeneous ODE, the way to solve this is to suppose a solution of the exponential type. This yields a second order polynomial equation (the "characteristic equation") that the exponent must satisfy. In case the solutions of the characteristic...
  38. C

    Solving ode with complex numbers

    I want to solve y''+y'+y=(sin(x))^2 and try to use y=Ae^{ix} but then when I square it I get A^2 e^{2ix} I found y' and y'' and solved for A and it didn't work I guess I could use the formula for reducing powers but I would like to try and get around that.
  39. MarkFL

    MHB Solving 2nd Order Homogeneous ODE - Joe's Question on Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  40. J

    Long-term behaviour of solution to ODE: oscillatory

    Homework Statement I wondered if anyone could advise me how to proceed with this question. The solution to the differential equation \frac{dQ}{dt}= \frac{1}{2}+\frac{1}{4}sin(t)-\frac{Q}{50} is Q=25+(\frac{sin(t)-625cos(t)+63150e^{-\frac{t}{50}}}{2501}) when Q_0= 50 "The long-term...
  41. J

    Initial Value problem 1st order ODE

    Homework Statement I have been trying to follow a solution to a problem I had but do not quite understand the whole thing. I wondered if anybody could clear it up for me. Let a_0 be the initial value of 'a' for which the transition from one type of behaviour to another occurs. The...
  42. MarkFL

    MHB Cassi's question at Yahoo Answers regarding a first order linear ODE

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  43. J

    Where did I go wrong with my solution to this first order ODE?

    Homework Statement Can anyone point out where I have gone wrong with this? Verify that the given function is a solution of the differential equation. y' -2ty =1 y= e^{t^2}\int^t_0 e^{-s^2}ds+e^{t^2} The Attempt at a Solution The steps I have taken are the following: i)...
  44. G

    Exact ODE and Finding Integrating Factors

    Homework Statement In my ODE class, we learned how to solve first order ordinary differential equations which are not exact yet but exact after multiplying by the right integrating factor. The integrating factor we learned about take one of the five forms: f(x), f(y), f(xy), f(x/y), and...
  45. E

    Solving Nonlinear ODE: magnetism with varying particle charge

    Hello. I have a set of ODE where 1) \frac{dv_x}{dt}=\frac{q(t)B}{m}v_y 2) \frac{dv_y}{dt}=\frac{q(t)B}{m}v_x 3) \frac{dv_z}{dt}=0 Following the strategy to solve a simple harmonic oscillator, I differentiate (1) to get 4) \frac{d^2v_x}{dt^2}=\frac{q(t)B}{m}\frac{dv_y}{dt}+q'(t)v_y...
  46. MarkFL

    MHB Bob's question at Yahoo Answers regarding equilibrum points of ODE

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  47. C

    Mathematica Solving ODE numerically in Mathematica - 'ndnum' error?

    I'm trying to solve this ODE R'(t)=\frac{-a}{R(t)^2} numerically in Mathematica (a, b are non-zero constants). Here's what I have: NDSolve[{R'[t]==-a/R[t]^2, R[0]==b, WhenEvent[R[t]==0, end=t; "StopIntegration"]}, R, {t,0,1}] It's returning with NDSolve:::ndnum : Encountered...
  48. W

    Solving 2nd ODE and Multivariable Calculus for Wave Equation

    Hello guys, I would like to ask some questions regarding my coursework, which is about 2nd ODE and multivariable calculus. Since we have the one-dimensional wave equation and values for the string stretched between x=0 and L=2: 0≤x≤L, t≥0 The string is fixed at both ends so we have ...
  49. AntSC

    Applying Boundary Conditions in Homogeneous ODE Problems

    Find the eigenvalues λ, and eigenfunctions u(x), associated with the following homogeneous ODE problem: {u}''\left ( x \right )+2{u}'\left ( x \right )+\lambda u\left ( x \right )=0\; ,\; \; u\left ( 0 \right )=u\left ( 1 \right )=0 Solution: Try u\left ( x \right )=Ae^{rx} , which gives...
  50. S

    Forward Euler Method for ODE system

    Homework Statement Solve the following system for 0<t<5 u^\prime = u-e^{-2t} v, u(0) = 1 v^\prime = u+3v, v(0) = -2 using Forward Euler method and implement the numerical scheme into a MATLAB code. Homework Equations Forward Euler : \vec x^(\prime)_{n+1} = \vec F(t,\vec x)...
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