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

    ODE Logistics Equation: Solving for Rabbit Population Growth Rate

    I have the following logistics problem that I am stuck about halfway thru: The time rate of change of a rabbit population P is proportional to the square root of P. At time t=0 (months) the population numbers 100 rabbits and is increasing at the rate of 20 rabbits per month. How many rabbits...
  2. C

    Solve Bernoulli ODE with Integrating Factor x^-4 and n=3: Am I Missing a Step?

    So I have the following Bernoulli ODE: x^2*y' + 2xy = 5y^3 I use an integrating factor of x^-4, my n value is 3. I am okay until I get to the very end, where I get y^-2 = (2+Cx^5)/x; the book shows y^2 = x/(2+Cx^5). Am I must missing an algebra step or did I make a mistake somewhere within?
  3. J

    Inhomogeneous second order ODE with non-constant coefficient

    Homework Statement Solve ODE of form y''+(2/x)y'=C*(e^y) where C is a constant Homework Equations The Attempt at a Solution I don't really see how to approach this one, so a point in the right direction would be great. Thanks,
  4. 2

    1st order homogenous ODE: (x+y)dy/dx=(x-y)

    Homework Statement Find a general solutionHomework Equations (x+y)\frac{dy}{dx} = x-y The Attempt at a Solution \frac{dy}{dx} = \frac{x-y}{x+y} let v=y/x y=xv \frac{dy}{dx} = v+x\frac{dv}{dx} now, v+x\frac{dv}{dx} = \frac{x-xv}{x+xv} = \frac{1-v}{1+v} = \frac{1}{1+v} -...
  5. C

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    I know that the rate of change with time of a population is proportional to the square root of t. T=0 is y = 100. Population increases at rate of 20 per month. I started out by trying to do dy/dt = p^.5. I am used to the population problems where I use y=Ce^(rt) but am having trouble...
  6. C

    Solve ODE Mixture Question: 100 Gal Water Tank

    Tank w/ 100 gal pure water. At time = 0, sugar/water mixture with .2 lbs of sugar per gallon enters at 3 gal/ minute. Drain opened at bottom allows sugar solution to leave at 3 gal per minute. Perfect (lol) mixing occurs. I show the following: Rate in: .2 x 3 = .6 Rate out: x/100 Volume: 100...
  7. M

    Understanding Symmetry in ODE Solutions

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  8. L

    Solving ODE Substitution: dy/dx= (4x sec(2y/x) +y) / x

    Homework Statement dy/dx= (4x sec(2y/x) +y) / x IC: y (1) = pi/4 Homework Equations The Attempt at a Solution So i can split that up into 4xsec(2y/x)/x +y/x = 4sec(2y/x) +y/x and let v= y/x dy/dx = xdv/dx +v 4sec(2v) +v = xv' +v 4sec(2v)=xv' which is...
  9. C

    Solving Bernoulli ODE | Step-by-Step Guide | Example with n=-2

    I am working on the following Bernoulli ODE: 3xy^2 y' = 3x^4 + y^3. I come up with n = -2, so v = y^3 and y' = (1/3)v^(-2/3) v'. My integrating factor was x^-1. I end up with y^3 = X^2 + Cx yet the book has the same thing except X^4 instead of X^2. That makes me think I'm going wrong with...
  10. C

    What is the best substitution for solving this ODE?

    So I am trying to figure out what substitution to use for the following ODE substitution: x^2*y' + 2xy = 5y^3; I initially moved the 2xy to the right but to no avail because when I tried to divide through by x^2 (to clear the left), I struggle to get the y/x format on the right. If the...
  11. P

    Error in Numerical Solution of ODE by Euler Method - Patrick

    Hi, I recently need to do some numerical simulation by Euler method to solve a PDE. However, I noticed that there are some errors which are obtained with bigger numerical steps, when applying Euler scheme. Since my major is not mathematics, I do not know what this phenomenon is called. I...
  12. J

    Finding the Homogenous Solution to a Variable Coefficients 2nd order ODE

    x y'' + (x + 1) y' = 2 x Solve for y(x). Due to the coefficients being a function of x, I have no idea where to start to find the homogenous solution (Complementary Function). I know how to proceed after this part with the variation of parameters method. I just have no idea where to...
  13. H

    First ODE (dc/dt the rate of change of chemical in a pond)

    lets say there is a pond of 1,000,000 gallons of water. and the total of 10,000g of chemical is evenly dissolved in the water Fresh water starts pouring into the pond at the rate of 300 gallons/hour and the pond also releases water at the rate of 300 gallons/hour (hence no change in total...
  14. chwala

    Convergence of ode & pde

    Hi, well let me put the question a bit clear...my concern area is on ode and pde...my question is when you solve a pde/ode analytically and get a solution by asymptotic means does this mean that if solution exists then ...when using convergence as an alternative way of getting solution of the...
  15. F

    Courses What is the difference between these two courses? ODE vs EDE?

    What is the difference between these two courses? ODE vs EDE? Hopefully this is the right place to ask. Does anyone know the difference between these two courses? And which one I should take? I'll be taking Linear Algebra (2nd semester of LA) with either one. I haven't decided which one to...
  16. O

    I have two ODE books, which first?

    My math background is the Calc 1-3 series, and an intro to ODE class, but honestly I don't remember much from the ODE class (i started having to work during the time i had the class and just wound up cramming for tests and going in). I was feeling the same way about Calc 1 & 2 ( i took them...
  17. L

    Solving a First Order ODE with y'(x)^2 = y^2 + xy and the Hint: u = y/x

    Homework Statement y'(x)^2 = y^2 +xy Homework Equations hint : let u = y/x The Attempt at a Solution I divide x^2 on both sides and end up with... y'= y^2/x^2 + y/x then using the hint, i get y' = u^2 + u but i do not know how to solve the DE from here.
  18. U

    MATLAB Troubleshooting ODE Systems in MATLAB: Common Errors and Solutions

    Hi to everyone, I have some problem in implementing a ODE system in matlab. function dC = Model(x,C) dC = zeros(2,1); dC(1) = -2/C(1) -3*dC(2); dC(2) = -3/C(2) -4*dC(1); [x,C] = ode23(@Model(x,C),[0 300],[56.9 0]); plot(x,C) The debugger says "? Input...
  19. T

    Second ODE - Using x = e^t show that the equation

    Homework Statement Show that the equation x= e^t converts the equation ax^{2}\frac{d^{2}y}{dx^{2}} + bx\frac{dy}{dx} + cy = 0 in which a,b,c are coefficients Homework Equations The Attempt at a Solution x = e^t and so does dx/dt. So you can write dx/dt = x using the chain rule dy/dx =...
  20. MathematicalPhysicist

    Understanding Riccati's ODE Variant: A Generalized RODE Explanation

    Any one knows how do you call an equation of the type: y' = q_0(x)+q_1(x) y+...+q_n(x) y^n Maybe generalized RODE?
  21. B

    Is There an Analytic Solution for This Crazy ODE?

    Can anyone tell me if this ODE has an analytic solution? And if it does, how the heck might I go about it? \left(\frac{1}{y^{2}}\frac{dy}{dx}\right)^{2}-\frac{A}{y^{3}}-\frac{B}{y^{2}}=D
  22. B

    Solve ODE System: x'=x+y², y'=-y

    Homework Statement Solve this system: x' = x + y² y' = -y The Attempt at a Solution My text solves this by guessing a particular solution. It says: For the second equation y' = -y yields y(t) = y_0(e^-t). Inserting thisinto the first equation, we must solve: x' = x +...
  23. J

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    Hi I need to find the solution of d^2y/dx^2 + 2x(dy/dx) = 0 I've solved it in Maple and get that y=a*erf(x)+b but I have no idea how to arrive at this answer! Any help would be great, thanks.
  24. C

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  25. C

    Estimating Growth Rate and Simulating World Population with ODEs | Matlab Help

    Homework Statement The rate of change of the population p is proportional to the existing population at any time t: dp/dt = k_g*p where k_g is the growth rate. The world population in millions from 1950 through 2000 was 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2555...
  26. D

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    (URGENT) Power series solution for ODE Homework Statement Supose there is an infinite series solution\sum b_{n}x^{n} for u''+4(x-(1/4))^2*u+C(x) = 0 where C(x) is a function (I get it in another problem, I'll put it in the relevant equations area), determinate the coefitients b_{0} b_{1}...
  27. P

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    Nonlinear 1st order ODE \frac{dH}{dt}=B-A*(H-Z)^{3/2} where: B,A and Z are known values H=f(t); H is function of t I've already solve this ODE numerically using a 4th order RK routine. But my question is, it is possible to get an analytical solution for H(t)?
  28. R

    Solving ODE with Data Points: Finding Equation and Integrating Method

    hey guys, i was given some data points and i had to find an equation to fit the model. now my differential equation is h'= ah^b - ch^d with b < 0. I can't find any method for integrating because i don't know the constants in the equation. but i have the data points so that must help somehow. i...
  29. G

    First Order ODE Help: Troubleshooting Tips for Solving Differential Equations

    Having a bit of trouble, what do i do next? Thanks.
  30. R

    Solving an ODE Related to Relativistic Mass Change

    Homework Statement \frac{dp}{dt} = \frac{d}{dt}\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}} = F Find v(t) show that v -> c as t -> infinity & find the distance traveled in time t if the object starts from rest. Homework Equations The Attempt at a Solution Ive tried rearranging it into either a...
  31. R

    Extension of Variation of Parameters to First Order Non-Linear ODE?

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  32. M

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    Homework Statement The Attempt at a Solution I did the "show that" part. But what is throwing me off is the x(0)=0 part. What is "x" a function of? Using the series in the square brackets, I found that when n=0, a_1 = a_0 ^2 n=1, a_2 = (a_1 * a_0)/2 n=2, a_3 = (a_0*a_2 + a_1^2 + a_2*a_0)/3...
  33. H

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  34. E

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  35. H

    Runge-Kutta for sovling 2nd ODE

    Hi, Could someone please show me how to solve the following simple problem using the Runge-Kutta (RK4) integration method? (tw')' + tw = 0 with w(0) = 1, w'(0) = 0 on the interval [0,1] by introducing the new variable v=tw' and considering the resulting first order differential system...
  36. H

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    Homework Statement (tw')' + tw = 0 with w(0) = 1, w'(0) = 0 on the interval [0,1] by introducing the new variable v=tw' and considering the resulting first order differential system involving w and v Report your computed solution (wh(1),vh(1)) for h=1/10. Homework Equations The...
  37. T

    Neutron Attenuation 1st order ODE Interpertation

    Hi, I'm having some trouble interpreting an equation. In Lamarsh's Introduction to Nuclear Engineering. The formulae for neutron attenuation is: I(x) = I_{0} exp(-\Sigma_{t} x I am given the formulae \frac{-dI}{I(x)} = \Sigma_{t} dx This formulae has been described as "the probability of a...
  38. R

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    Hey guys, I am in dire need of help. In physics today, my professor gave us an experiment. It involved a strip of paper towel with dots along it from a sharp. it was hung from a metal ring stand and the edge of the paper towel was immersed into a dish of water. Water traveled up the paper towel...
  39. H

    How to Model Fish Population Dynamics Using First Order Differential Equations?

    The initial mass of fish in a lake was 7 thousand pounds on January 1st, 2001. Since the time, there was a 4-year moratorium on the harvesting on this specific type of fish. This species of fish reproduce at a rate proportional to the mass and by next year on the same date, there were 11.54...
  40. C

    Registration: Calc III, ODE, Physics 2, Statics & Engineering

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  41. P

    Bessel function Solution to Second order ODE with exponential coefficient

    Homework Statement Find the general solution to x'' + e^(-2t)x = 0, where '' = d2/dt2 Homework Equations - The Attempt at a Solution First I did a change of variables: Let u = e^(-t) Then du/dt = -e^(-t) dx/dt = dx/du*du/dt = -e^(-t)*dx/du d2x/dt2 = d/du(dx/dt)du/dt =...
  42. M

    Wronskian; second order ODE

    y1 and y2 are solutions to the ODE L[y]=0=y''+p(x)y'+q(x)yWhat can you conclude about p(x), q(x) and the solutions on the interval I if i) W(x) = 0 for all X on I ii) W(x) = c for all X on I, c =/= 0 --- W(x) = y_1'y_2-y_1y_2' = C*e^{\int{p(x)}} i) W=0 so y1'y2=y1y2' And y1 and y2 are...
  43. A

    Chasing Ducks: Solving a Differential Equation in a Square Pen

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  44. R

    ODE w/ Discontinuous forcing function

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  45. D

    Second Order ODE with Weird Coefficients: Solving the Equation x2-2x+1

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  46. L

    How to solve x^2 y' + 2xy = arctan(x) - ODE

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  47. J

    How can I convert a 2nd order ODE to a 1st order ODE?

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  48. G

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    Every second order linear ODE a(x)y''+b(x)y'+c(x)y=F(x) can be transformed into the forms; u''+g(x)u=h(x) divide whole equation to a(x) and choose y(x)=u(x)^(e^s(x)) but how i choose s(x) ?Please help...
  49. F

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  50. T

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