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. U

    Construct ODE that approaches an asymptote

    Homework Statement Construct a first order linear differential equation whose solutions have the required behavior as t approaches infinity. Then solve your equation and confirm that the solutions do indeed have the specified property. All solutions are asymptotic to the line y = 2 - t as t...
  2. 1

    Solve Linear ODE with Discontinuous f(x)

    Homework Statement $$\frac{dy}{dx}+y=\left\{\begin{matrix}1, \ 0\leq x< 1 \\ 0, \ x\ge1 \ \ \ \ \ \ \ \end{matrix}\right.$$ Homework Equations The Attempt at a Solution $$P(x)=1$$ Integrating factor ##=e^{x}## For ##f(x)=1##: $$\frac{d}{dx}[e^{x}y]=e^{x}$$ Integrating...
  3. I

    Integrating Factor for Solving ODE with Linear Coefficients

    Problem: xy'+2y=3x Attempt: Divide by x... y'+\frac{2y}{x}=3 I think I find the integrating factor by doing: e^{\int \frac{2}{x}dx} Not sure if that's right but if it is then the solution to the integral is just 2x. Any help is appreciated
  4. 1

    Solve Linear ODE Using Integrating Factor

    Homework Statement Solve the initial value problem: $$sin(x)y' + ycos(x) = xsin(x), y(2)= \pi/2$$ Homework Equations The Attempt at a Solution Recognizing it as a Linear First-Order Equation:$$\frac{dy}{dx}+y\frac{cosx}{sinx}=x$$ $$P(x)=\frac{cosx}{sinx}$$ Integrating...
  5. T

    Solving a First ODE Using an Integrating Factor

    $$w'+2w=0\\ \frac { dw }{ dx } =-2w\\ I(x)={ e }^{ 2x }\\ \frac { dw }{ dx } { e }^{ 2x }=-2w{ e }^{ 2x }\\ \int { \frac { dw }{ dx } { e }^{ 2x } } dx=\int { -2w{ e }^{ 2x } } dx$$ Not sure what to do next.
  6. 1

    Solve Linear ODE Simplify Step

    Homework Statement Solve: ##x\frac{dy}{dx}-4y=x^{6}e^{x}## Homework Equations ##x^{-4}\frac{dy}{dx}-4x^{-5}y=xe^{x}## is equal to ##\frac{d}{dx}[x^{-4}y]=xe^x## The Attempt at a Solution The second equation above simplifies to the third (according to my textbook) but I can't figure out...
  7. MarkFL

    MHB Verifying a Solution to an ODE: Differential Equations HW Help!

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  8. T

    Reducing a PDE to an ODE Using a Change of Coordinates

    I've been studying Walter A. Strauss' Partial Differential Equations, 2nd edition in an attempt to prepare for my upcoming class on Partial Differential Equations but this problem has me stumped. I feel like it should be fairly simple, but I just can't get it. 10. Solve ##u_{x} + u_{y} + u =...
  9. A

    Question on Ordinary Differential Equation (ODE)

    Homework Statement Find the ODE of the following (1) du/dy = -u (2) d^2u/dxdy = -du/dx Homework Equations For question 1, the answer is u= A(x)e^(-y) while for question 2, the answer is u= e^(-y)(B(X) + c(Y)) The Attempt at a Solution I've already solved the question, but...
  10. T

    2nd Order ODE Contradiction ?

    2nd Order ODE "Contradiction"? To solve a 2nd order ODE, we can follow the steps as shown below. (Image 2 is a continuation from Image 1, apologies for the size difference.) :rolleyes: The method to obtain the solution is straightforward. Let's say \frac{d^2y}{dx^2}=ky If k = -1, a...
  11. F

    Mass on a spring non-homogeneous second order ODE

    Homework Statement A mass of 5kg stretches a spring 10cm. The mass is acted upon by an external force of 10sin(t/2) Newtons and moves in a medium that imparts a viscous force of 2N when the speed of the mass is 4cm/sec. If the mass is set in motion from its equilibrium position with an initial...
  12. MarkFL

    MHB Lauren's question at Yahoo Answers regarding inexact ODE

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  13. P

    Can't decide between PDE or ODE or both

    Hey everyone I am going to be a freshman this fall (in college). I am currently having a dilemma in choosing my math class. In high school I took classes all the way up to Honors Differential Equations (ODE). In June I went to the university and signed up for Ordinary Differential Equation...
  14. C

    Matching Inner and Outer Expansions for Approximating ODE Solutions

    I'm trying to approximate f'(r) for the following equation using matched asymptotic expansions -\frac{1}{2}\epsilon ff''=\left[\left(\epsilon+2r\right)f''\right]' where \epsilon \ll 1 and with the boundary conditions f(0)=f'(0)=0, \quad f'(\infty)=1 The inner expansion which satisfies...
  15. C

    How can I find an outer expansion for f'(r) in this ODE?

    Ahoy! I'm trying to approximate f'(r) for the following equation using matched asymptotic expansions -\frac{1}{2}\epsilon ff''=\left[\left(\epsilon+2r\right)f''\right]' where \epsilon \ll 1 and with the boundary conditions f(0)=f'(0)=0, \quad f'(\infty)=1 The inner expansion which...
  16. D

    2nd Order ODE Initial Value Proof Problem

    [b]1. Check that y(t)=1/λ ∫_0-t_〖f(s) *sin(λ(t-s) )ds〗 is the solution of the following initial value problem y''(t)+λ^2y(t)=f(t), λ>0, y(0)=0,y'(0)=0 Homework Equations [b]3. I tried to do integration by parts on y(t), but...
  17. W

    Struggling with ODE: Find Particular Solution y(0)=1

    i am having issues solving an ODE it is given as y'= (1-2y-4x)/(1+y+2x) I've been told to find the particular solution when y(0)=1 please help
  18. T

    Solve second order ode with Green function

    I had made a post in the past about the same problem and unfortunately I wasn't clear enough so I am trying again. I am studying an article and there I found without any proof that the solution of: Let ##g \in \mathbb{C}## and let ##u:(0,\infty)\to \mathbb{C}## $$ -u''+\lambda^2u=f\,\, on...
  19. T

    How can I solve a coupled PDE and ODE using the method of lines?

    I am trying to solve an ODE and PDE and I am having problems coming up with a method for doing so. The PDE is: k1*(dC/dt) = k2*(dC/dx) But I have an ODE that is an expression for dC/dt: dC/dt = k3*C Where k1,k2 and k3 are constants. I planned to use the method of lines to get...
  20. F

    Physics nonlinear ODE example for numerical methods project?

    I am doing a little research project into numerical methods of solving ODEs where I do 1 half of learning about the basics of numerical methods and then look at a particular method (Linear multistep) and then the second half is looking at a particular example, applying what I've learned and...
  21. T

    Solve second order ode with Green's functions

    -u''(z)+α2u(z)=f(z), u(0)=g(z), u(z)=0 as z→∞ -u''(z)+α2u'(z)=f(z), u(0)=g(z), u(z)=0 as z→∞ I am interested to solve these two boundary problems using Green's functions. It is noticed that z is complex variable. Can someone help me to do this?
  22. MarkFL

    MHB Paul's question at Yahoo Answers regarding a 3rd order linear homogeneous ODE

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  23. I

    Solving a nonhomogeneous 2nd order ode

    Hi, everyone! This is my first post here, I need an hand with this equation! Homework Statement Solve the initial value problem: \begin{equation} \begin{cases} u''(x)+4u(x)=\cos(2x) \\u(0)=u'(0)=1 \end{cases} \end{equation} The Attempt at a Solution I started by solving the...
  24. F

    Feedback control via ODE variable coefficients?

    If one has a simple variable coefficient process like y'(t) = r(t)*y(t), is there a way to control it to a set point by feedback hitting the variable coefficients in r(t)? I am interested in feedback control of population processes. y'(t) = r*y(t) is simple proportional growth with a...
  25. M

    ODE with 2 parameterized families

    Homework Statement problem: Find a 1-parameter family of solutions of each of the following equa- tions. Assume in each case that the coefficient of dy \neq 0. (x + \sqrt{ y^2 - xy}) \mathop{dy} - y \mathop{dx} = 0 answer: y = ce^{-2\sqrt{1 - x/y}}, \;\;\; y >0, \, x< y; \;\;\; y =...
  26. R

    Connection between numerical integration and solving ODE numerically.

    Hey I have taken a programming course. And I have learned about Simpson, Trapezoidal and the midpoint rule etc, I have programmed these. I have also implemented forward Euler, backward euler, Runge Kutta etc for solving ODE. I am wondering if there is any way to unify these two things, are...
  27. S

    Solving 2nd Order ODE: y'' + 2y' - y = e^{-x}, y(0) = y'(0) = 1

    Homework Statement Consider the following second order ODE $$ y'' + 2y' - y = e^{-x}, \quad y(0) = y'(0) = 1. $$ Convert this to a system of first order equations and use the pc33assisys MATLAB file to compute the solution for y(2). Homework Equations The Attempt at a Solution...
  28. I

    How Do You Solve an ODE Involving Changing Tank Volume and Concentration?

    I am trying to solve this ODE and am stuck on this step! It is a mass balance of a tank where the volume and concentration are changing by time Fin*Co - Fout*C1 = d(C1*V)/dt Fin*Co - Fout*C1 = d(C1)/dt * V + d(V)/dt * C1 where V = A*h (area and height, where area is constant and height...
  29. MarkFL

    MHB Marcus Right's question at Yahoo Answers regarding a first order homogeneus ODE

    Here is the question: Here is a link to the question: Homogenous Differential Help with equation? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  30. L

    Solving second-order ODE with Runge-Kutta 4

    Homework Statement Note: I think there is a typo here but I'm not sure. Is there supposed to be a comma between the delta t/2 and y_n on K2 and K3, and delta t and y_n on K4? Homework Equations See above.The Attempt at a Solution Substituting dy/t = z gives \frac{dz}{dt} = 3z - 2ty - cos(t)...
  31. F

    How can I solve a first-order non-linear ODE?

    Hi! I'm having a lot of trouble solving the following ODE: dx/dt = A - B*sin(x) where A and B are constants. My ODE skills are a bit rusty, and I wasn't able to find anything on the Internet that could help me, so could someone please show me how to solve for x in terms of t? I've...
  32. O

    Periodic Solution to FIrst Order ODE Proof

    Homework Statement Consider the differential equation x' = f(t,x) where f(t,x) is continuously differentiable in t and x. Suppose that f(t+T,x) = f(t,x) for all t Suppose there are constants p, q such that f(t,p) > 0, f(t,q) < 0 for all t. Prove that there is a periodic solution...
  33. E

    About taking ODE directly after taking 1 variable calculus

    This is my first time posting in this forum, I am not very familiar with the rules. I am a year 1 physics student, and I had only taken 1 variable calculus( I also know some basic linear algebra, e.g. how to calculate eigenvector in 2x2 situation, don't know Gram–Schmidt process for...
  34. Kudaros

    MATLAB Droplet Profile in Matlab- ODE stability

    Hello, I'm currently modeling the profile of a droplet (sessile drop, axisymmetric) in matlab. I've coded differential equations, applied the solver, and I get a reasonable result, except that it spirals continuously. The ODE's in question are: \frac{dx}{ds}=cos(\theta)...
  35. MarkFL

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

    Here is the question: Here is a link to the question: General solution of dy/dt=k((y)(b-y))? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  36. E

    MATLAB Stability issues of ODE solutions using Matlab

    Hello, guys I am struggling with attaining stability values for u in solving the diffusion equation. The stability of u depends on the value of r from : D=1000; r0=1000; std=1.0; tau=1.0; IP=2500; %initial pressure % % Radial grid and inhomogeneous term nr=51; dr=r0/(nr-1)...
  37. K

    Solving the Ode Emden-Fowler Equation: A Scientist's Perspective

    Hi, y'' = y^2/x y(x) = ? Trial and error: y(x) = 2/x. I am glad to get a particular solution too. Thanks, kamke
  38. S

    Solving 2nd ODE for RLC circuit

    This is really more of a mathematical question than physics. Given a RLC circuit, I will arrive at the following DE: \ddot{Q}+\frac{R}{L}\dot{Q}+\frac{1}{LC}Q-\frac{\epsilon}{L}=0 How do I solve for Q(t)??
  39. R

    Solution for higher order wave ODE

    Hi guys, Here is an equation that I have tried for few days to solve and still haven't succeeded, I'm interested to solve this 4th order wave equation to find u(x). ∫∫(A u(x) + B u(x)2 + C u(x)3 +D u''(x)) dx dx=0 the 4th term is second derivative of displacement u(x). I assume...
  40. MarkFL

    MHB Logan's question at Yahoo Answers involving an IVP with a linear 1st order ODE

    Here is the question: Here is a link to the question: Initial value problem? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  41. L

    ODE application of damped motion

    Homework Statement A mass of 40 g stretches a spring 10 cm. A damping device imparts a resistance to motion numerically equal to 560 (measured in dynes/(cm/s)) times the instantaneous velocity. Find the equation of motion if the mass is released from the equilibrium position with a downward...
  42. L

    Applications of ODE; damped motion

    Homework Statement A force of 2 lb stretches a spring 1 ft. A 3.2 lb weight is attached to the spring and the system is then immersed in a medium that imparts a damping force numerically equal to 0.4 times the instantaneous velocity. Find the equation of motion if the weight is released from...
  43. D

    Integrating factor for a 2nd order homogeneous linear ODE

    Homework Statement Consider the general linear homogeneous second order equation: P(x)y'' + Q(x)y' + R(x)y = 0 (1) We seek an integrating factor μ(x) such that, upon multiplying Eq. (1) by μ(x), we can write the resulting equation in the form [μ(x)P(x)y']' + μ(x)R(x)y = 0...
  44. djh101

    Physical Chemistry Electives: ODE, Mechanics, or Mathematical Methods?

    Hello, everyone. I am currently a junior [physical] chemistry major and am picking out my future upper division electives. I've narrowed them down to a handful of classes and what I'm looking for is just a little background information on them, which ones might be better than others, general...
  45. r-soy

    MHB Show whether the given ODE is exact, then solve

    I am given the following ODE, and the instructions are to show whether it is exact or not, and then solve: (x+y)dy=(y-x)dx My first step, is to put the equation in the form M(x,y)\,dx+N(x,y)\,dy=0: (x-y)dx+(x+y)dy=0 Next, I compute the partials: \frac{\delta M}{\delta y}=-1\ne1=\frac{\delta...
  46. S

    Solving an ODE about a point using a solution about another point?

    Homework Statement The first task was to solve ##(1-x)y''+y=0## about x = 0, which I've already found. Now I have to use this solution to solve ##\color{red}{xy''+y=0}## about x = 1. Homework Equations The Attempt at a Solution I found the solution about x = 0 (after a lot of...
  47. E

    MATLAB Can't run my main Matlab file related to ODE solution

    Hey, guys Can you please help me to spot mistakes in numerical solution of following diffusivty equation: ∂P/∂t= 0.001127*k/(μ*ϕ*c_t )*((∂P/∂x)^2*c+(∂^2 P)/(∂x^2 )). Matlab give the following command: Undefined function or variable 'r'. Error in function_handle2 (line 9) for...
  48. S

    Solving an ODE with power series method

    Homework Statement Solve ##(1-x)y''+y=0## at the point ##x_0=0##. Use this solution to find a solution to ##xy''+y=0## around the point ##x_0=1##. Homework Equations The Attempt at a Solution ##(1-x)y''+y=0## ##(x-1)y''=y## ##\displaystyle\sum_{k=2}^\infty a_k k...
  49. M

    Series Solution of an ODE: Finding a Non-Recursive Formula

    Homework Statement Solve for y' = x^2y The Attempt at a Solution There's something that's been really bothering me about this question and similar ones. We assume that the solution to the ODE will take the form y = \sum_{n=0}{a_nx^n} After finding y', plugging in the expressions...
  50. N

    Using Laplace Transforms to Solve ODE with Piecewise Forcing Function

    Homework Statement ODE: y'' + 4y' + 3y = f(t) f(t) = (?? HELP - What's the mathematical term to describe these? I can't seem t o find it in my notes :cry: ) 1, 0 ≤ t < 2 t², 2 ≤ t < 3 0, t ≥ 3 Write a brief description on how you would solve this ODE using Laplace transforms. Also use the...
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