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. Another

    Problem ODE from mechanics equation

    The particles are moving under force F = rK when r = 1/u and u = 1/r I tried to solve the problem by defining a variable v = u4 and u = v1/2 But I can not divide variable v out of variable u I want to find r (Radius) of orbit
  2. M

    Solving the Implicit Euler ODE with Boundary Conditions

    Homework Statement Write an implicit Euler code to solve the system ##c'(x) = \epsilon c''(x)-kc(x)## subject to ##1-c(0)+\epsilon c'(0) = 0## and ##c'(1)=0##. Homework Equations Nothing out of the ordinary comes to mind. The Attempt at a Solution In the following code, there is central...
  3. B

    Single-integral solution to 2nd order inhomogeneous ODE

    Homework Statement I want to show that $$f''(x) = g(x)$$ has a solution of the form $$f(x) = 2\int_0^{x} dx' (x-x') g(x').$$ It's not hard to verify that it is a solution, the question is how to find it. This should be easy and is likely a standard problem but I haven't found the right...
  4. M

    Solution to Asymptotic ODE System for Small ε: x and y Expressions

    Homework Statement Find through ##O(\epsilon)##, for ##\epsilon \ll 1##, the solution to the system $$\epsilon x'(t) = -x+y\\ \epsilon y'(t) = -(\epsilon+1)y+x$$ Homework Equations ##x = \sum x_n\epsilon^n## and ##y = \sum y_n\epsilon^n## The Attempt at a Solution Substituting the...
  5. Another

    Question ODE non-homogeneous Linear

    Mod note: Member warned that the homework template is NOT optional find yp (particular integral) (D2 + 4D + 5) y = 2 e-2xcos(x) ((D+2)(D+2)+1) y = 2 e-2xcos(x) yp = [1/((D+2)2 + 1)] ⋅ 2e-2xcos(x) yp = e(-3)x ∫ ∫...
  6. M

    Matched Asymptotic ODE Solution for ##\epsilon d_x(xd_xf)-xf=0##

    Homework Statement Solve to order ##\epsilon## $$\epsilon d_x(xd_xf)-xf=0$$ subject to ##|f(0)|<\infty## and ##f(1)=1## via matched asymptotic expansions. Homework Equations Nothing comes to mind. The Attempt at a Solution Perform a matched asymptotic analysis. In this case when I take a...
  7. W

    Solving Boundary Value ODE: y''+λy=0

    Homework Statement ## y''+\lambda y = 0 ; y(0) = 0, y(\pi)-y'(\pi) = 0## Homework EquationsThe Attempt at a Solution So, we have to test when lambda is equal to, less than and greater than 0. Let ## \lambda = 0## thus, the ODE becomes ## y'' = 0 ## which implies solutions of the form ## y(t) =...
  8. W

    Boundary Value ODE: Eigenvalues & Functions

    Homework Statement Find the eigenvalues and eigenfunctions of the following boundary-value problem. ## y''+\lambda y = 0 ; y(0) = 0, y'(L) = 0 ## Homework EquationsThe Attempt at a Solution So, we have to test when lambda is equal to, less than and greater than 0. Let ## \lambda = 0## thus...
  9. M

    Estimating Eigenvalues from linear ODE

    Homework Statement Given $$u''(x)+\lambda u = 0\\ u(-1)=u(1)=0.$$ If ##\lambda_0## is the lowest eigenvalue, show that ##4 \lambda_0 = \pi^2##. Homework Equations $$\lambda_0 = glb\frac{(L(u),u)}{(u,u)}$$ where ##glb## denotes greatest lower bound and ##L## is the Sturm-Louiville operator. I...
  10. S

    Write 2nd order ODE as system of two 1st order ODEs

    Homework Statement Write the following second-order ODE as a system of two first-order ODEs. ##d^2y/dt^2 + 5(dy/dt)^2 - 6y + e^{sin(t)} = 0## Homework Equations w = dy/dt The Attempt at a Solution The solution of the book says ##dy/dt = w, dw/dt = -5w - 6y + e^{sin(t)}##, but shouldn't it be...
  11. C

    Second order(?) ODE + Runge-Kutta method question

    Homework Statement When a rocket launches, it burns fuel at a constant rate of (kg/s) as it accelerates, maintaining a constant thrust of T. The weight of the rocket, including fuel is 1200 kg (including 900 kg of fuel). So, the mass of the rocket changes as it accelerates: m(t) = 1200 - m_ft...
  12. B

    Linear ordinary differential equation.

    Homework Statement ##\dfrac{dy}{dx} + y = f(x)## ##f(x) = \begin{cases} 2 \qquad x \in [0, 1) \\ 0 \qquad x \ge 1 \end{cases}## ##y(0) = 0## Homework EquationsThe Attempt at a Solution Integrating factor is ##e^x## ##e^x\dfrac{dy}{dx} + e^x y = e^x f(x)## ##\displaystyle ye^x = \int e^x...
  13. B

    Solving Homework Equations for x & y with 2x + 3y + 1 = 0

    Homework Statement ##(2x + 3y + 1)dx + (4x + 6y + 1) dy = 0## ##y(-2) = 2## Homework EquationsThe Attempt at a Solution Let ##z = 2x + 3y## then ##z^\prime = 2 + 3y^prime## ##\displaystyle \dfrac{(z + 1)}{2z + 1} + \dfrac13\left({dz \over dx} - 2\right) = 0## ##\dfrac{dz}{dx} = \dfrac{z-...
  14. S

    I Constructing a 2nd order homogenous DE given fundamental solution

    Homework Statement Given a set of fundamental solutions {ex*sinx*cosx, ex*cos(2x)} Homework Equations y''+p(x)y'+q(x)=0 det W(y1,y2) =Ce-∫p(x)dx The Attempt at a Solution I took the determinant of the matrix to get e2x[cos(2x)cosxsinx-2sin(2x)sinxcosx-cos(2x)sinxcosx-...
  15. Derp215

    Admissions Reapplying to Engineering: Physics II, Calc II Or ODE?

    Hello, I just came across this forum and thought of asking for advice! I am reapplying to Engineering and I am taking summer school (Full term) and need to take 2 more classes to boost my avg. Right now, I am enrolled in Physics II, Calculus III, and ODE and I have to choose two out of the...
  16. karush

    MHB -17.2.02 - Solve 2nd order ODE using undetermined coefficients.

    $\tiny{17.2.02}$ \nmh{1000} $\textrm{Solve the equation by the method of undetermined coefficients.}\\$ \begin{align*}\displaystyle y''-4y'&=\sin{x}\\ y_p&=A\sin{x}\\ \end{align*} $\textit{ answer}$ \begin{align*}\displaystyle y{\left (x \right )}& = C_{1} + C_{2} e^{4 x} - \frac{1}{17}...
  17. whatisgoingon

    Second order ODE into a system of first order ODEs

    Homework Statement The harmonic oscillator's equation of motion is: x'' + 2βx' + ω02x = f with the forcing of the form f(t) = f0sin(ωt)The Attempt at a Solution So I got: X1 = x X1' = x' = X2 X2 = x' X2' = x'' ∴ X2' = -2βX2 - ω02X1 + sin(ωt) The function f(t) is making me doubt this answer...
  18. Poetria

    Finding Solutions for Second Order ODE with Initial Condition y(0)=6

    Homework Statement How many functions y(t) satisfy both y''+t^2*y=0 and y(0)=6? 2. The attempt at a solution As this is a second order differential equation, two initial conditions (for y and y') would be needed to obtain a unique solution (cf. existence and uniqueness theorem). So the...
  19. M

    Properties of Solutions of Matrix ODEs

    Homework Statement We assume from ODE theory that given a smooth A: I → gl(n;R) there exists a unique smooth solution F : I → gl(n;R), defined on the same interval I on which A is defined, of the initial value problem F' = FA and F(t0) = F0 ∈ gl(n;R) given.(i) Show that two solutions Fi : I →...
  20. M

    How Do Matrix ODEs Relate to Determinants and Traces?

    Homework Statement Please bear with the length of this post, I'm taking it one step at a time starting with i) Let A: I → gl(n, R) be a smooth function where I ⊂ R is an interval and gl(n, R) denotes the vector space of all n × n matrices. (i) If F : I → gl(n, R) satisfies the matrix ODE F'...
  21. Poetria

    Basis for the space of solutions (ODE)

    Homework Statement The equation given: dy/dt = 3*y A basis for the space of solutions is required.The Attempt at a Solution According to me it is e^(3*t) but it has turned out false. Why? I am considering the answer "The basis is the set of all functions of the form c*e^(3*t) but a...
  22. Sirsh

    4DOF Spur Gear System - Eigenvalues not corresponding with the Eqns?

    Hi there, I am modelling a four degree of freedom system which is the dynamics of two spur gears in mesh, having two rotational and two translation degrees of freedom, respectively, a diagram exhibiting the system can be seen below. I have derived the equations of motion (EOM) and...
  23. M

    Runge Kutta to solve higher order ODE

    Homework Statement Edit* should say F'(0) = F(0) = 0 Homework Equations I know that I typically need 3 equations for a 3rd order ODE, does this apply if the is no F'? In the picture above are the equations I came up with, am I on the right trail? Lastly I am familiar with RK4, however I have...
  24. M

    Solution to complex valued ODE

    Homework Statement Let f : I → C be a smooth complex valued function and t0 ∈ I fixed. (i) Show that the initial value problem z'(t) = f(t)z(t) z(t0) = z0 ∈ C has the unique solution z(t) = z0exp(∫f(s)ds) (where the integral runs from t0 to t. Hint : for uniqueness let w(t) be another...
  25. Euler2718

    (Ordinary) Differential Equation Trouble

    Homework Statement Find the solution of the differential equation by using appropriate method: t^{2}y^{\prime} + 2ty - y^{3} = 0 Homework Equations I'm thinking substitution method of a Bernoulli equation: v = y^{1-n} The Attempt at a Solution [/B] t^{2}y^{\prime} + 2ty - y^{3} = 0...
  26. A

    B First Order Non-Linear ODE (what method to use?)

    Hi, The problem is to solve: dy/dx = −[2x + ln(y)]*(y/x) Attempt: I have tried to see if it is exact, I found it not to be, I can't easily find a function to multiply by to make it exact either (unless I am missing something obvious). It clearly isn't seperable, nor is it homogenous (I know...
  27. H

    I How to find the integrating factor? (1st order ODE)

    x2 + y + y2dx - x dy = 0 Integrating factor, I(x,y) = -1 / (x2 + y2) How to find the integrating factor ?Why I cannot use below method to solve the ode ? (1/N)(My - Nx) = g(x) , I(x,y)=exp( ∫ g(x) dx) OR (1/M)(My - Nx) = h(y) , I(x,y)=exp( -∫ h(y) dy)
  28. M

    I Analyzing a 2nd Order Non-Linear ODE with Variable Substitution

    Can someone check my work here? Both ##f=f(x)## and ##y=y(x)##. $$f'y'+\frac{fy''}{1+y'^2}=0\implies\\ \frac{y''}{y'(1+y'^2)}=-\frac{f'}{f}\\ \frac{y''}{y'(1+y'^2)}=-\ln(f)$$ Now let ##v=y'##, which implies $$...
  29. S

    First order separable Equation ODE

    Homework Statement \frac{dy}{dx}\:+\:ycosx\:=\:5cosx I get two solutions for y however only one of them is correct according to my online homework (see attempt at solution) Homework Equations y(0) = 7 is initial condition The Attempt at a Solution \int \:\frac{1}{5-y}dy\:=\:\int...
  30. L

    Linearly-damped rotational motion

    http://imgur.com/a/8QjoW http://imgur.com/a/8QjoW Hello- I am trying to determine the dynamics of this linearly-damped hinge. Assuming that: v(0) = 0 damping constant = b door has mass = m I was able to determine that: ∑Fx = -Fd * cos(45-θ/2) + Rx = m*dvx/dt ΣFy = -Fd * sin(45-θ/2) - Fg +...
  31. C

    I Solution of an ODE in series Frobenius method

    Hi I am supposed to find solution of $$xy''+y'+xy=0$$ but i am left with reversing this equation. i am studying solution of a differential equation by series now and I cannot reverse a series in the form of: $$ J(x)=1-\frac{1}{x^2} +\frac{3x^4}{32} - \frac{5x^6}{576} ...$$ $$...
  32. Euler2718

    I Can Nonlinear Equations be Linearized Using Free Parameters?

    I am given the equations of Lorenz with respect to deterministic non-periodic flow: \frac{dX}{dt} = Pr(Y-X), X(0)=X_{0} \frac{dY}{dt} = -XZ + rX - Y, Y(0) = Y_{0} \frac{dZ}{dt} = XY-bZ, Z(0) = Z_{0} where Pr is the Prandtl number, r = Ra/Rac is the ratio of the Rayleigh number to its...
  33. Cocoleia

    How to find r(t) when we are given conditions - ODE

    Homework Statement Consider the following problems In #2, they start the solution by saying: r(t)=u(t-1) in #3, they start by saying that r(t)=t-tu(t-1) I understand how to solve the problem once you get r(t), I just don't understand how they decide what r(t) is going to be.
  34. Cocoleia

    Second order non homogeneous ODE, IVP

    Homework Statement I need to solve: x^2y''-4xy'+6y=x^3, x>0, y(1)=3, y'(1)=9 Homework EquationsThe Attempt at a Solution I know that the answer is: y=x^2+2x^3+x^3lnx Where did I go wrong. I was wondering if it's even logical to solve it as an Euler Cauchy and then use variation of parameters...
  35. Kanashii

    4th order RK to solve 2nd order ODE

    Homework Statement Consider the initial value problem x" + x′ t+ 3x = t; x(0) = 1, x′(0) = 2 Convert this problem to a system of two first order equations and determine approximate values of the solution at t=0.5 and t=1.0 using the 4th Order Runge-Kutta Method with h=0.1. Homework Equations...
  36. W

    2nd Order Linear ODE-Derivation of system-issue

    Homework Statement How exactly they combined equation1 and equation2 and got that system? I don't get that part. Homework Equations A*(dy/dt)= -k*y eq1 A*(dz/dt)=ky-kz eq2 The Attempt at a Solution I tried substituting the 1st ky in the 2nd equation and then differentiating but I don't...
  37. RJLiberator

    ODE: System of Linear Equations usuing Diff. Operator

    Homework Statement This is an ordinary differential equation using the differential operator. Given the system: d^2x/dt - x + d^2y/dt^2 + y = 0 and dx/dt + 2x + dy/dt + 2y = 0 find x and y equation Answer: x = 5ce^(-2t) y = -3ce^(-2t) Homework EquationsThe Attempt at a Solution We change...
  38. Telemachus

    Numerical methods for a system of coupled ODE

    Hi there. I have to solve a system of coupled ordinary differential equations. I have some initial values, but in different points of the domain. The equations are all first order. Let's suppose the system looks like this: ##\displaystyle\frac{dy_1}{dz}=y_1+y_2+0.01##...
  39. Euler2718

    Rewriting ODE's into lower orders

    Homework Statement Express \frac{d^{2}x}{dt^{2}} + \sin(x) = 0 In a system in terms of x' and y'. Homework EquationsThe Attempt at a Solution [/B] I seen this example: x^{\prime\prime\prime} = x^{\prime}(t)\cdot x(t) - 2t(x^{\prime\prime}(t))^{2} Where they then wrote: x^{\prime} =...
  40. E

    I Can you check the solution for this second order ODE?

    The second order ODE is, \begin{equation*} \frac{d^2 x}{dt^2} = -\omega^2_g \frac{dx}{dt} \end{equation*} I tried solving this by substitution of the second order derivative into a variable and transforming the equation into a second order polynomial, and I get the solution involving an...
  41. S

    B Can This ODE Solution f'(x)=f(2-x) Be Correctly Solved?

    Hello! I have this problem f'(x)=f(2-x) and I need to find f. This is what I did x -> 2-x f'(2-x)=f(2-2+x)=f(x) => f''(x)=f'(2-x)=f(x) => f''(x)=f(x) => ##f(x)=c_1e^x+c_2e^{-1}##. So, ##c_1e^x-c_2e^{-x}=c_1e^{2-x}+c_2e^{x-2}## => ##-c_2e^{-x}=c_1e^2e^{-x}## => ##-c_2=c_1e^2##. And, similarly...
  42. mr.tea

    I Constant solution and uniqueness of separable differential eq

    Hi, I am learning ODE and I have some problems that confuse me. In the textbook I am reading, it explains that if we have a separable ODE: ##x'=h(t)g(x(t))## then ##x=k## is the only constant solution iff ##x## is a root of ##g##. Moreover, it says "all other non-constant solutions are separated...
  43. M

    Obtaining General Solution of ODE

    Homework Statement So they want me to obtain the general solution for this ODE. Homework Equations I have managed to turn it into d^2y/dx^2=(y/x)^2. The Attempt at a Solution My question is, can I simply make d^2y/dx^2 into (dy/dx)^2, cancel the power of 2 from both sides of the equation...
  44. Kanashii

    Solve for the solution of the differential equation

    Homework Statement Solve for the solution of the differential equation and use the method of variation of parameters. x`` - x = (e^t) + t Homework Equations [/B] W= (y2`y1)-(y2y1`) v1 = integral of ( g(t) (y1) ) / W v2 = integral of ( g(t) (y2) ) / W The Attempt at a Solution [/B] yc= c1...
  45. Valour549

    I 2nd Order Linear ODE w/ nonconstant coefficient

    xy'' + 2xy' - y = 0 Honestly no clue where to start, Wolfram Alpha gives a rather complex answer lol (http://www.wolframalpha.com/input/?i=xy%27%27%2B2xy%27-y%3D0)
  46. RJLiberator

    ODE: Solving using Laplace Transform

    Homework Statement Solve: y''+λ^2y = cos(λt), y(0) = 1, y'(π/λ) = 1 where t > 0 Homework EquationsThe Attempt at a Solution I start off by taking the Laplace transform of both sides. I get: L(y) = \frac{s}{(s^2+λ^2)^2}+\frac{sy(0)}{s^2+λ^2}+\frac{y'(0)} {s^2+λ^2} Now take the inverse...
  47. M

    ODE homogeneous equations w/constant coefficients

    Homework Statement Find the general solution y"+3y'+2y=0 Homework Equations y(t) =c_1e^r_1t + c_2e^r_2t The Attempt at a Solution a=1 b=3 c=2 r^2+3r+2=0 (r+2)(r+1)=0 r_1=-2 r_2=-1 General solution: y(t) =c_1e^(-2t)+c_2e^(-t)I was wondering if the order mattered. The answer in the book is...
  48. A

    I Interval of existence and uniqueness of a separable 1st ODE

    Problem: y'=((x-1)/(x^2))*(y^2) , y(1)=1 . Find solutions satisfying the initial condition, and determine the intervals where they exist and where they are unique. Attempt at solution: Let f(x,y)=((x-1)/(x^2))*(y^2), which is continuous near any (x0,y0) provided x0≠0 so a solution with y(x0)=y0...
  49. RJLiberator

    Shifting Origin Method for Solving ODEs

    Homework Statement Find a particular solution to: (3x+2y+3)dx - (x+2y-1)dy = 0, y(-2) = 1 The answer to this problem as presented in the book ODE by Tenenbaum is the following: (2x+2y+1)(3x-2y+9)^4=-1. Homework Equations I will be shifting the origin to try to compute this problem. The...
  50. N

    MHB Solving a Non-Exact ODE: $(y-2x^2y)dx +xdy = 0$

    Solve the ode $$(y-2x^2y)dx +xdy = 0$$ The equation is in exact form $$Q(x,y)dx+ P(x,y)dy =0$$ When I test for exactness it fails. Then I used the technique $$\frac{M_y-N_x}{N}$$ I get $u(x)=-2x$ as my integrating factor. But I end still end up with a non-exact d.e why is that...
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