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.
The set of equations is
I have first tried to solve only first two equations (removing the components of other 4 equations from them.)
This is the output, where first column is the time, 2nd - X_p, and 3rd - X_n
In case of an integral ##\rightarrow## differential equation of the type:
$$ f(t) = \int_0^t g(f(\tau)) d\tau $$
$$ \rightarrow \frac{df(t)}{dt} = g(f(t)) $$
which turns out not to be solvable in exact form because ##g(f(t))## is a non-polynomial function (but it would if ##g(f(t))## was a...
Hello! I am trying to solve the time dependent Schrodinger equation for a 2x2 system and I ended up with this ODE:
$$y''=-iA\sin{(\omega t)}y'-B^2y$$
with the initial conditions ##y(t=0)=0## and ##y'(t=0)=B##. I can look at it numerically but I was wondering if there is a way to get something...
I'm aware that I can introduce the perimeter p = dy/dx
then I can rearrange my equation to make y the subject, then I can show that dp/dx = p/x. However, this only gives me a bunch of quadratic curves for my solution. However given part b I see that two curves are meant to intersect each point...
My code is as follows: but when I use the function in my command window exactsol(t) and input a tolerance but there is an error in LINE 19 saying unrecognized ivpfun, could someone help me fix it as I am unsure of how to proceed from here.
function y = exactsol(t)
y = zeros (2,1);
y(1) =...
My approach is to use the definition of the Force with ##\displaystyle F = \frac{dp}{dt} = \dot{m} v + m \dot{v}##. Since ##m(t)## decreases linearly, I should be able to set ##m(t) = M - \Phi t##, thus ##F = - \Phi v + (M - \Phi t) \dot{v}##, which gives ##\displaystyle v = -\frac{ F - (M -...
The characteristic equation ## m^3 -6m^2 + 12m -8 = 0## has just one single, I mean all three are equal, root ##m=2##. So, one of the particular solution is ##y_1 = e^{2x}##. How can we find the other two? The technique ##y_2 = u(x) e^{2x}## doesn't seem to work, and even if it were to work how...
I am trying to solve this two level (Schrodinger) equation as a function of time:
$$i\begin{pmatrix}
\dot{x}\\
\dot{y}
\end{pmatrix} = \begin{pmatrix}
0 & iW+dE_0sin(\omega t)\\
-iW+dE_0sin(\omega t) & \Delta
\end{pmatrix}\begin{pmatrix}
x\\
y
\end{pmatrix}$$
(I can go into more details about...
Hi all,
I have another second order ODE that I need help with simplifying/solving:
##p''(x) - D\frac{e^{\gamma x}}{A-Ae^{\gamma x}}p'(x) - Fp(x) = 0##
where ##\gamma,A,F## can all be assumed to be nonzero real numbers and ##D## is a purely nonzero imaginary number.
Any help would be appreciated!
Hello,
I'm posting here since what follows is not about homework, but constitutes a personal research which underlies some more general questions.
As with the infamous "casus irreducibilis" (i.e. finding the real roots of a cubic function sometimes requires intermediate calculations with...
Hello,
I am currently planning on self studying math analysis with MIT ocw courses, but I cannot find the analysis course. I have found on the web that analysis and ODE are the same thing. If I want to get a good introduction to analysis, is the MIT ODE course fit for such a use?
Thanks
I believe I am doing everything right up until the point where I have to try and find a recurrence relation. I honestly have no idea what to do from there. I've listed my work in getting the powers of n and the indicies to all match. Any help appreciated.
Here is the original DE...
I believe it is the case that any linear second order ode with at most 3 regular singular points can be transformed into a hypergeometric function. I am trying to solve the following equation for a(x):
where E, m, v, k_{y} are all constants and I believe turning it into hypergeometric form will...
Hey all,
I am currently struggling decoupling (or just solving) a system of coupled ODEs.
The general form I wish to solve is:
a'(x)=f(x)a(x)+i*g(x)b(x)
b'(x)=i*h(x)a(x)+j(x)a(x)
where the ' indicates a derivative with respect to x, i is just the imaginary i, and f(x), g(x), h(x), and j(x) are...
Hi PF!
Given the following ODE $$(p(x)y')' + q(x)y = 0$$ where ##p(x) = 1-x^2## and ##q(x) = 2-1/(1-x^2)## subject to $$y'(a) + \sec(a)\tan(a)y(a) = 0$$ and $$|y(b)| < \infty,$$ where ##a = \sqrt{1-\cos^2\alpha} : \alpha \in (0,\pi)## and ##b = 1##, what is the Green's function?
This is the...
Hello!
Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For...
Ignoring the second part of the question for now, since I think it will be more clear once I understand how this equation is homogeneous.
According to my textbook and online resources a first-order ODE is homogeneous when it can be written like so:
$$M(x,y) dx + N(x,y) dy = 0$$
and ##M(x,y)##...
I'm studying ODEs and have understood most of the results of the first chapter of my ODE book, this is still bothers me. Suppose
$$\begin{cases}
f \in \mathcal{C}(\mathbb{R}) \\
\dot{x} = f(x) \\
x(0) = 0 \\
f(0) = 0 \\
\end{cases}.
$$
Then,
$$
\lim_{\varepsilon \searrow...
Equation:
, where matrix D, C, G and F can be represented by
I'm supposed to design a control system that looks like this:
I am given that the dynamic model = fcn(D,C,G,dq) where the dq is the same as 𝑞̇ and d2q in the diagram is the same 𝑞̈. The default initial value of [𝑞(0), 𝑞̇(0)] is...
Here is my paper. A criticism and other comments are welcome.
Abstract: The Lagrange-D'Alembert Principle is one of the fundamental tools of classical mechanics. We generalize this principle to mechanics-like ODE in Banach spaces.
As an application we discuss geodesics in infinite dimensional...
I am trying to find a way to prove that a certain first order ode has a unique
solution on the interval (1,infinity). Usually the way to do this is to show that
if x' = f(t,x) (derivative with respect to t), then f(t,x) and the partial derivative with respect to f are continuous.
However, this...
this is pretty easy for me to solve, no doubt on that. My question is on the constant. Alternatively, is it correct to have,
##ln x= \frac {v^2}{2}##+ C, then work it from there...
secondly, we are 'making" ##c= ln k##, is it for convenience purposes?, supposing i left the constant as it is...
i am new to MATLAB and and as shown below I have a second order differential equation M*u''+K*u=F(t) where M is the mass matrix and K is the stifness matrix and u is the displacement.
and i have to write a code for MATLAB using ODE45 to get a solution for u. there was not so much information on...
function Asteroid_Mining
clc
%Initial conditions
g0 = 9.81; %gravity (m/s^2)
p = 1.225; %atmospheric density at sea level (kg/m3)
Re = 6378; %radius of Earth (km)
Ra = 7.431e7; %distance of Bennu from Earth in (km) [August 2023]
G = 6.674e-11/1e9; % Gravitational constant (km3/kg.s2)
mu =...
Using MATLAB, I am trying to calculate the time taken for a spacecraft to travel from Earth to a near Earth asteroid and then returning back to Earth but so far I have had no luck. Furthermore, I want to plot a Hohmann transfer and calculate the mass of fuel required for this mission. If...
I am trying to self study Ordinary Differential Equations and am totally fed up of "cookbook style ODEs". I have recently finished Hubbard's Multivariable Calculus Book and Strang's Linear algebra book. I would like a rigorous and Comprehensive book on ODEs. I have shortlisted a few books below...
Non-homegenous first order ODE so start with an integrating factor ##\mu##
$$\mu=\textrm{exp}\left(\int a dt\right)=e^t.$$
Then rewrite the original equation as
$$\frac{d}{dt}\mu y = \mu g(t).$$
Using definite integrals and splitting the integration across the two cases,
$$\begin{align}...
Sorry the problem is a bit long to read. thank you to anyone who comments.
We consider the initial value problem for the Burger's equation with viscosity given by
$$\begin{cases} \partial_t u-\partial^2_xu+u\partial_xu=0 & \text{in}\quad (1,T)\times R\\\quad \quad \quad \quad \quad...
I have following differential equation dV/dt = 5 - 2 * V(t)^(1/3) which represents a the time its take to drain a barrel of rain water which contain 25 Liter of water, at t = 0.
I am suppose to calculate the least amount of water in barrel during this process.
If I set the rate of growth to...
The ODE given to us is y' = xcosy. I am having a bit of trouble when it comes to solving this problem. We are supposed to show that the solution has a local minimum at x = 0 with the hint to think of the first derivative test. However, I am only really familiar with the first derivative test...
I have an ODE, which is y' = 5x3(y-1)1/5 with the initial condition y(0)=1, I must find two solutions.
My attempt at solving this problem is as follows:
Separate the equation, we get, dy/(y-1)1/5 = 5x3dx.
Integrate both sides, ∫ dy/(y-1)1/5 = ∫ 5x3dx.
We are left with, (5/4)(y-1)4/5 = (5x4)/4...
I'm new to learning about ODE's and I just want to make sure I am on the right track and understanding everything properly.
We have our ODE which is y' = 6x3(y-1)1/6 with y(x0)=y0.
I know that existence means that if f is continuous on an open rectangle that contains (x0, y0) then the IVP has...
This is a very simple question: I would like to solve for ##\psi## in this equation $$\frac{d^{2}\psi}{d\xi^2} =\xi^2\psi$$
I so apply ##y=c_{1}e^{-kx}+c_{2}e^{kx}## and ##\psi## should be equal to ##\psi=c_{1}e^{-\xi^2}+c_{2}e^{\xi^2}##, because ##(D^2-\xi^2)\psi=0##. However the answer is...
Hello,
I am trying to compute some non-linear equations with pseudospectral/collocation methods. Basically I am expanding the solution as
$$
y(x)=\sum_{n=0}^{N-1} a_n T_n(x),
$$
Being the basis an Chebyshev polynomial with the mapping x in [0,inf].
Then we put this into a general...
Say you have the set of coupled, non-linear ODEs as derived in this thread, it has two unknowns ##N(t)## and ##\theta(t)##:
$$ N - mg = - m\frac{L}{2}\left(\dot{\theta}^2\cos(\theta) + \ddot{\theta}\sin(\theta)\right)$$
$$ \frac{L}{2}N\sin(\theta) = \frac{1}{12}ml^2\ddot{\theta}$$
What freedom...
to
I am a bit clueless on how to get break the ##r X(r)## from inside the derivative.
P.S. I tried to copy from Symbolab instead of pasting the picture, but it didn't let me.
I'm looking for a general analytical solution to a particular ODE that comes up in neuroscience a lot. My feeling is that such a solution can't be obtained, otherwise someone would have presented it by now, but I don't have a good understanding of why it is so hard to solve.
The equations as...
I'm trying to find the local truncation error of the autonomous ODE: fx/ft = f(x).
I know that the error is |x(t1) − x1|, but I can't successfully figure out the Taylor expansion to get to the answer, which I believe is O(h^3).
Any help would be greatly appreciated!
Actually I was trying to write a small program in Scilab to simulate a quantum particle. When I give a potential higher than energy, the wave function should go like exp(-x) and decay. But my program just increases without bound.
Is there any nice way to do anything about it?
Question:
So I got around on doing this example, and I'm pretty sure I messed up somewhere, would appreciate if someone could point out what I did wrongly.
1) For any second ODE, I should let:
##y_{1}= y ## and ##y_{2}= y' ##
Hence,
##y_{1}'= y' = y_{2} ## and ##y_{2}'= y'' = xy(x)+x^2-y(x) =...
With the new variable, I got:
$$p^2 (p'_y)^{2}=k^2(1+p^2)$$ where ##p'_y## is ##\frac{dp}{dy}##.
I modified the equation so the variable p and dp can be separated from dy. Here what I got:
$$\frac{p}{\sqrt{p^2+1}} dp=k dy$$
I substitute ##p^2+1=u## so I got
$$\sqrt{u}=ky+c_1$$
Back substitution...
This equation, is non linear, non-separable, and weird. I would like to have a direction to start working on this.
I tried writing sin(2y) = 2sin(y)*cos(y).
See,
##xy' = x^3sin^2(y)-2sin(y)cos(y)##
Can't separate.
Writing in this way:
##(x^3sin^2y-sin2y)dx-xdy=0##
Also, I checked that it is...
Suppose I'm solving
$$y''(t) = x''(t)$$ where $$x(t)$$ is the ramp function. Then, by taking the Laplace transform of both sides, I need to know $x'(0)$ which is discontinuous. What is the appropriate technique to use here?
Greetings,
I have a question to the following section of the book https://www.springer.com/gp/book/9783319163741:
I understand that the equation is separable, since I can just write
$$ \int_{x_0}^{x} \frac {1}{V(x', \xi, \eta)}dx' =\int_{0}^{t}dt' .$$
However, without knowing the exact shape...
Homework Statement:: ODE -> Transfer Function Assistance
Relevant Equations:: Newtonian physics, buoyancy, drag
[Mentor Note -- thread moved to DE from the schoolwork forums, since it is for work and not schoolwork]
Hello all,
I'm new here but I'm looking for a bit of guidance with a...