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. ognik

    MHB Are There Complex Solutions for a 2nd Order ODE?

    Hi - I have y''+9y=0 which is a constant coefficients ODE, the CE is then $r^2+9=0$ and I get a general solution $ y=C_1e^{3ix}+C_2e^{-3ix} $ But I have seem these solutions written as ACos3x+BSin3x. If I use Euler on my solution, I get $ C_1(Cos3x + iSin3x) +C_2(Cos3x-iSin3x) $ ... Are they...
  2. MidgetDwarf

    Arnold ODE. What is the intended audience?

    Greetings, what is the intended audience of Arnold's ODE book? I have completed a computational style ODE course, 3 semesters of Calculus from Stewart's book and a Linear Algebra Course. Will I need higher mathematics in order to study Arnold's book on ODE?
  3. H

    MATLAB Matlab ode solvers - adding a separate convergence criteria

    Hi, is there a way to add a user defined convergence criteria to an ode solver so that the solution is stopped?
  4. J

    Analysis ODE by V.I Arnold (doubts about publishers)

    Hi everyone! I'm thinking of buying Arnold's book of ODE, but looking for it on the internet I have found two publishers which have this book printed: MIT Press and Springer. The main question is: which edition is better? Both editions haven't the same quantity of pages, maybe because of...
  5. K

    2nd order ODE circuit, switch question

    if we assume the condition immediately after switch is closed (t=0+), *Capacitor voltage cannot jump. *Inductor current cannot jump. dv(0+)/dt=i_c(0+)/c di(0+)/dt=v_l(0+)/L which means we can find the initial condition of the post-zero system algebraically. However, it contradcits to the...
  6. T

    Solve Highly Nonlinear ODE: Fourier Series Analysis

    I have an equation of motion given by $$f(z(t),t) = \frac{d^2z}{dt^2} + A\frac{dz}{dt} + B $$ where $$f(z(t),t) = [(\frac{C}{z^2+C^2})^2-(\frac{D}{z^2+D^2})^4]^2(1+cos(wt))$$ and ##A,B,C,D,## and ##w## are constants Is it possible to solve this for ##z(t)##? I have been solving it numerically...
  7. dumbdumNotSmart

    Finding the homogenous solution of Var.Coeff. 2nd Order ODE

    It's been too long guys. I've given this ODE lots of thought and still no cigar. Homework Statement We are given the following ODE: $$ (x-a)y''-xy'+a^2y = a(x-1)^2e^x $$ and knowing that y=e^x is a solution to the homogenous equation, find the possible values of a. Next part: Using the...
  8. I

    MHB How to solve an ODE in powers of x?

    I have this question due soon and I have no idea how to do it. Please help me get started on it Solve in powers of x: (1-x^2)y''-2xy'+42y=0
  9. H

    Finding general solution of an ode using substitution

    Homework Statement By making the transformation u= x^αy where α is a constant to be found, find the general solution of[/B] y'' + (2/x)y' + 9y=0The Attempt at a Solution I've worked out y,y',y'' and subbed them into get x^-au'' + x^a-1(2-2a)u' + x^-a-2(x^2-a(a-1))u =0...
  10. Tzabcan

    Find General Solution for Inhomogeneous 2nd ODE

    If i have 3y" - 2y' -y = 14 + e2x+8x And i want to find the general solution. Obviously first i obtain the characteristic eqn, yc, by making it into a homogeneous eqn. Then i can get yp BUT Am i able to get yp for the e2x and the 14 + 8x separately, then add them together for yp?Thanks
  11. C

    Non-linear non-constant coefficient second order ODE

    I would like to solve the steady-state one dimensional heat equation for a two piece material system. The thermal conductivity in each segment is a linear function of temperature, where ##\kappa_1=a_1T+b_1## for material 1 and ##\kappa_2=a_2T+b_2## for material 2. ##a_1, a_2, b_1, and \;b_2##...
  12. W

    Series solution of ODE near singular points with trig

    Homework Statement Given the differential equation (\sin x)y'' + xy' + (x - \frac{1}{2})y = 0 a) Determine all the regular singular points of the equation b) Determine the indicial equation corresponding to each regular point c) Determine the form of the two linearly independent solutions...
  13. T

    System of ODEs with RK4 & step doubling in Fortran : damping

    Hello, I'm recently trying to code a solver for a system of differential equations u'(t) = F(t,u), using a Runge Kutta 4 method with an adaptative stepsize. For this, I'm using the 'step doubling' method, which is the following : suppose that we now the solution u(i) at time t(i). Then, the...
  14. P

    Solve a 2nd order Ordinary Differential Equation

    Homework Statement Y''-((Y')^2)+(C1*exp(Y))=C2 C1 and C2 are constants. exp = e Homework Equations No clue how to start this The Attempt at a Solution Y'=A=dY/dt Y=At+C3 (not sure) A'-(A^2)+C1exp(At+C3)-C2=0 A'-(A^2)+C1exp(C3)exp(At)=0 let C=C1*exp(C3) A'-(A^2)+Cexp(At)=0
  15. N

    Find the approximate linear ODE system

    dx/dt = x-y^2 dy/dt= x^2 -xy -2x For each critical point, find the approximate linear OD system that is valid in a small neighborhood of it. I found the critical points which are (0,0),(4,2),(4,-2) but have no idea how to do the above question! please help!
  16. B

    Estimating the Parameter 'a' from ODE System

    Hi everyone I have a system of ODE as follows x1_dot=f1(t)-ax1 x2_dot=f2(t)-ax2 x3_dot=f3(t)-ax3 f1,2,3(t) are unknown nonlinear functions of time, a is constant and unknown, x1,2,3 and their derivatives are given. How can I estimate the parameter a from the given information? Thanks
  17. A

    Optimizing Savings Growth with Continuous Compounding and Depletion

    Homework Statement Find the amount in a savings account after one year if the initial balance in the account was $1000, the interest is paid continuously into the account at a normal rate of 10% per annum (compounded continuously), and the account is being continuously depleted at the rate of...
  18. Mark Brewer

    Finding a Second Linearly Independent Solution Using Reduction of Order

    Homework Statement (Reduction of order) The function y1 = x-1/2cosx is one solution to the differential equation x2y" + xy' + (x2 - 1/4) = 0. Use the method of reduction of order to find another linearly independent solution. The Attempt at a Solution I divided x2 to both sides to get the...
  19. Y

    Solving ODEs with Heun's Method: A Question

    Hi there, in my notes for Heun's method for solving an ODE, I have y(new) = y(old) + 0.5(k1 + k2)Δh And k1 is supposed to be f(y(old)) while k2 is f(y(old) + q11k1Δh) and q11 is 1 So if for example I have a simple differential equation like du/dt = au It would be du/dt = 0.5(k1 + k2) du/dt...
  20. P

    Matrix-free iteration methods and implicit ODE solvers

    Im trying to implement the implicit Euler method in high-performance software for micromagnetic simulations, where I'm restricted to using the driving function of the ODE (Landau-Lifshitz equation) and the previous solution points. This obviously not a problem for an explicit method, since we...
  21. I

    Solving ODE with Laplace transform

    Homework Statement Let ##f(t)= \begin{cases} \sin t , \; \; 0 \le t < \pi \\ 0 , \; \; \; \; \; \text{else.} \end{cases}## Use Laplace transform to solve the initial value problem ##x'(t)+x(t)=f(t), \; \; \; x(0)=0.## Homework Equations Some useful Laplace transforms...
  22. M

    Write an Equation given a solution to an ODE

    Homework Statement Give an example of a system of differential equations for which ##(t,1)## is a solution. Homework Equations Nothing comes to mind. The Attempt at a Solution I thought to initial pose the system as an eigenvalue problem ##\vec{x}' = A \vec{x}##. However, ##(t,1)## is...
  23. W

    Construct a second order ODE given the solutions?

    Homework Statement I've been stuck on this problem for three days now, and I have no clue how to solve it. Construct a linear differential equation of order 2, for which { y_1(x) = sin(x), y_2(x) = xsin(x)} is a set of fundamental solutions on I = (0,\pi) . Homework Equations Wronskian for...
  24. SteliosVas

    Second Order Non linear ode matlab

    Homework Statement Okay the problem is of a free swinging pendulum with dampening which is modeled using the following equation: Damping coefficient: c=1 s−1 Mass: m=1 kg Gravity: g=9.81 ms−1 Link length: l=0.5 m We know θ(0)=90° and θ′(0)=0, solve this equation from t = 0 to t = 10...
  25. D

    System of ODE - comparison with paper

    I have the following system of differential equations, for the functions ##A(r)## and ##B(r)##: ##A'-\frac{m}{r}A=(\epsilon+1)B## and ##-B' -\frac{m+1}{r}B=(\epsilon-1)A## ##m## and ##\epsilon## are constants, with ##\epsilon<1##. The functions ##A## and ##B## are the two components of a...
  26. B

    Physical interpretation for system of ODE

    If an ODE of 2nd order like this A y''(x) + B y'(x) + C y(x) = 0 has how physical/electrical interpretation a RLC circuit, so, how is the electrical interpretation of a system of ODE of 1nd and 2nd order? \begin{bmatrix} \frac{d x}{dt}\\ \frac{d y}{dt} \end{bmatrix} = \begin{bmatrix}...
  27. B

    RLC, RL, RC and LC circuits and ODE

    Look this schematic picture: S means source and can be a current source or a voltage source. C_1, C_2 and C_3 are linear components, can be resistor, capacitor and inductor. In the everycircuit, I tried set up all possible combination and the everycircuit denied some combinations, but...
  28. ognik

    MHB Please check ODE operator solutions

    1) Given $\mathcal{L}u=0$ and $ g\mathcal{L}u$ is self-adjoint, show that for the adjoint operator $ \bar {\mathcal{L}}, \bar{\mathcal{L}}(gu)=0$ Is it enough to say that if self-adjoint, then $ \mathcal{L}= \bar {\mathcal{L}} $. I assume g represents a function of x (so no inner products with...
  29. ognik

    MHB Why is there a * in the 2nd integral for self-adjoint ODEs?

    Hi, my textbook claims $ <u|\mathcal{L}v> =\int_{a}^{b}u^*\mathcal{L}v \,dx = \int_{a}^{b} u(p_0u''+p_1u'+p2u) \,dx$, u,v matrices or functions My only query is why $u^*$, and where did the * get to in the 2nd integral? I am used to $ <f|g>=\int_{a}^{b}f(x)g(x) \,dx $ ...
  30. applestrudle

    How do you solve this 2nd ODE for a pendulums displacement...

    ..when it is released from rest with velocity (v0, 0) I can get 1.6.5 but I can't get this:
  31. K

    Understanding Integrals and the Importance of Constants

    Homework Statement Suppose an RC series circuit has a variable resistor. If the resistance at time t is given by by R = a + bt, where a and b are known positive constants then the charge q(t) on the capacitor satisfies (a+bt) q' + (1/C)q = V where V is some constant. Also q(0) = q_0 Find q(t)...
  32. gfd43tg

    ODE 45 with coupled ODE's in a matrix, reactor temp.

    Homework Statement My question is regarding part (e), I just gave all the questions for reference. Homework EquationsThe Attempt at a Solution These are the coupled equations I should solve (from part d) My issue is using ode45 to get ##C_{A}(t)##, ##C_{P}(t)##, and ##T(t)##. Here is my...
  33. T

    Zero-Input/Zero-State Response vs. Homogenous/Particular Solution

    I have a question regarding the solutions to linear-ordinary differential equations. I had originally learned that the solutions to such differential equations consist of a homogenous solution and particular solution. The homogenous response is due to initial conditions while the particular...
  34. bayan

    Solve 3 Coupled ODEs with NDSolve - Error Fix & Alternatives

    Hi, I need to solve 3 coupled first order ODE's using NDSolve (numerical solution). This is the code I have used ; NDSolve[{u'[t] == ((1 - (u[t]/t^2))/(-3 - (u[t]/t^2) - (v[t]/t^2))), v'[t] == (((u[t]/t^2) - (v[t]/t^2))/(-3 - (u[t]/t^2) - (v[t]/t^2))), w'[t] == (v[t])/(-3 t^2 - (u[t]) -...
  35. Mark Brewer

    Ordinary Differential Equations (ODE) Problem

    Homework Statement dy/dx + 2sin2pix = 0 -------Answer: y = 1/pi cos2pix + c Homework EquationsThe Attempt at a Solution I made several attempts but no success to the correct answer. The first step I made was subtracting 2sin2pix to both sides. I then used integration by parts, and this is...
  36. ognik

    MHB Is This a Generalized Form of Legendre's Differential Equation?

    The ODE is $ \d{}{x}[(1-x^2)\d{u}{x}]+\alpha u + \beta x^2u = 0 $. I know of Legendre's ODE, and Bessles and a few others - does this one also have a name?
  37. ognik

    MHB Use Wronskian to show only 2 indepndent solutions of 2nd order ODE

    Given standard ODE $ y'' + P(x)y' + Q(x)y=0 $, use wronskian to show it cannot have 3 independent sltns. Assume a 3rd solution and show W vanishes for all x. so 1st row of W = {$ {y}_{1}, {y}_{2},{y}_{3} $}, 2nd row is 1st derivatives, 3rd row is 2nd derivatives. I can find the determinate...
  38. S

    Solving an ODE using shooting method

    Homework Statement I have been trying to solve the following nonlinear ordinary differential equation: ##-\Phi''-\frac{3}{r}\Phi'+\Phi-\frac{3}{2}\Phi^{2}+\frac{\alpha}{2}\Phi^{3}=0## with boundary conditions ##\Phi'(0)=0,\Phi(\infty)=0.##Homework Equations My solution is supposed to...
  39. ognik

    MHB Help with some fiddly algebra around a 1st order ODE

    Frustratingly although I can solve the ODE, I am getting a different answer to the book. Now going in circles so would appreciate a fresh pair of eyes. The ODE (for a boat coasting with resistance proportional to $V^n$) starts as $ m\frac{dV}{dt} =-kV^n $ Find V(t) and x(t), V(0) = $V_0$ I...
  40. A

    Trapping region for a nonlinear ODE system

    I need to find a trapping region for the next nonlinear ODE system $u'=-u+v*u^2$ $v'=b-v*u^2$ for $b>0$. What theory i need to use or which code in Mathematica o Matlab could help me to find the optimal trapping region.
  41. B

    Reducing and increase of order and ODE

    Two questions: First: If is possible to reduce the order of an ODE increasing the number of equations, so, is possible do the inverse patch? In other words, is possible reduce the number of equations of a system of ODE increasing the order? Second: This technique of reducing and increasing of...
  42. B

    How to reduce a system of second order ODEs to four first order equations?

    Someone can explain me how to get the general solution for this system of ODE of second order with constant coeficients: \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix} \begin{bmatrix} \frac{d^2x}{dt^2}\\ \frac{d^2y}{dt^2}\\ \end{bmatrix} + \begin{bmatrix} b_{11} &...
  43. L

    Intro Math Mastering Differential Equations

    During the summer, I plan on learning differential equations (ODE's and PDE's) from bottom to top, but I am unable to choose books due to a great variety present. Can you suggest books for me to read in the following order (you can add as many books in each section if you like);Ordinary...
  44. N

    Numerical methods that need a guess/approximate solutions

    Hello everyone! I am currently playing with an old analog computer, which could solve time-dependent ODE/PDEs pretty fast, without time-stepping. But the problem with analog computer's solutions is that they are not very accurate. I am very curious that is there any numerical method/solver which...
  45. Remixex

    Hooke's and Newton's law to find Second order ODE

    Homework Statement A weight of 8 pounds extends a spring 2 feet. It's assumed that the damping force that acts on the system is equal (number-wise) to alpha times the speed of the weight. Determine the value of alpha > zero so x(t) is critically damped. Determine x(t) if the weight is liberated...
  46. popopopd

    Solution space of nth order linear ODE, n dimension Vector Space

    if we do picard's iteration of nth order linear ODE in the vector form, we can show that nth order linear ODE's solution exists. (5) (17) example) (21) (22) (http://ghebook.blogspot.ca/2011/10/differential-equation.html)I found that without n number of initial conditions, the solution...
  47. M

    Solving Non-Linear ODE: Tips and Guidance from PF Community

    Hi PF! Can any of you help me reduce this ODE to find a solution? $$y y''+2y'^2+xy'+\frac{1}{2}y = 0 \implies \\ (y y')'+y'^2+xy'+\frac{1}{2}y = 0 \implies\\ (yy')'+(xy)'+y'^2-\frac{1}{2}y=0$$ but here I am stopped. Am I even going the correct route? I know I can re-write this equation as...
  48. W

    Solving First Order Linear ODE: dy/dx = y/x + tan(y/x)

    Homework Statement Solve dy/dx = y/x + tan(y/x) Homework EquationsThe Attempt at a Solution Not separable, as far as I can tell. It's not homogeneous, since for the tan term f(λx,λy) = tan(λy/λx) = tan(y/x) ≠ λtan(y/x). It's also not of the form dy/dx + P(x)y = Q(x), because I don't think Q(x)...
  49. popopopd

    Nth order linear ode, why do we have n general solutions?

    hi, I looked up the existence and uniqueness of nth order linear ode and I grasped the idea of them, but still kind of confused why we get n numbers of general solutions.
  50. mlouky

    Fortran I to solve ODE with rk4 y" + 2y =0 in fortran

    How can i solve y" + 2y =0 with RK4 and their program fortran please I néed your helpe
Back
Top