For a book included in some editions of the Septuagint, see The Book of Odes.The Odes of Solomon is a collection of 42 odes attributed to Solomon. Various scholars have dated the composition of these religious poems to anywhere in the range of the first three centuries AD. The original language of the Odes is thought to have been either Greek or Syriac, and to be generally Christian in background.
Let ## \mathbf{x''} = A\mathbf{x} ## be a homogenous second order system of linear differential equations where
##
A = \begin{bmatrix}
a & b\\
c & d
\end{bmatrix}
## and ##
\mathbf{x} = \begin{bmatrix}
x(t)\\ y(t))
\end{bmatrix}
##
Now to solve this equation we transform it into a 4x4...
If I have been given a system of inhomogeneous linear ODEs,
$$
\vec{x'} =
\begin{bmatrix}
4 & -1 \\
5 & -2 \\
\end{bmatrix}
\vec{x}
+
\begin{bmatrix}
18e^{2t} \\
30e^{2t}\\
\end{bmatrix}
$$
I have found its particular solution to be:
$$
1/4
\begin{bmatrix}
-31e^{2t} - 25e^{6t} \\
85e^{2t} -...
Hello :
Trying to find references on drawing direction fields of higher order differential equation by hand as 1st step then by computer , do you know any reference I can read ( PDF , books ,...) , and hope it is not only some short notes
Best regards
HB
The Coupled ODE Model
Below are my coupled differential equations, where the only variable I try to meddle with is the ITMblood. The motivation here is if I try to increase ITMblood (in the next section I will show how I do it), at some concentration of ITMblood (most likely a very huge one) ...
I am trying to solve the following set of 5 first order ODEs
Where the variables are ##\Theta_0, \Theta_1, \Phi, \delta## and ##v##. The initial conditions (ICs) are
(Note that there is a typo in the above ICs; it is ##v## instead of ##u##). I am following this solved sample, following...
I use the following script and function in MatLab, but get three errors.
I shall first write down the code and after that the errors that I get.
function yprime = lorenz_de(t,y)
%LORENZ_DE Lorenz equations.
% yprime = lorenz_de(t,y).
yprime = [10*(y(2)-y(1))...
I know how to solve ODEs using both methods. The problem I'm having is knowing when to use one and not the other. If someone could help clarify this for me. I can't find the correct section in my textbook.
given the differential equation
$\quad y''+5y'+6y=0$
(a)convert into a system of first order (homogeneous) differential equation
(b)solve the system.
ok just look at an example the first step would be
$\quad u=y'$
then
$\quad u'+5u+6=0$
so far perhaps?
Summary: Looking for guidance on how to model an RLC circuit with a system of ODES, where the variables are the resistor and inductor voltages.
This is a maths problem I have to complete for homework.
The problem is trying to prove that the attached circuit diagram can be modeled using the...
Hello,
I'm having a problem with this system. Ignore the physics.
I have the feeling it should be tremendously easy... but I can't figure it out.
I don't know how to extract it from the pdf so I'll post just the these 2 pages.
https://ufile.io/39ovq
The equations are (1.14) and (1.15), the...
Whilst studying symplectic integrators (as a hobby!) I accidentally stumbled on http://www.maia.ub.edu/~angel/taylor/taylor.pdf, which contains a link to GPL source code for the method described. I found it fascinating, especially since searching around the topic (Taylor Series Methods)...
Homework Statement
Attached
Homework Equations
Below
The Attempt at a Solution
To be honest I was going to differentiate one equation to get a 2nd order ODE and plug in the other equation, since to me ##v(0)=0## is not strong enough to do as below, am I completely mis-interpreted...
Hi,
I am looking at the following system of ODEs:
\begin{eqnarray*}
\dot{\omega}_{3}+\alpha\omega_{3} & = & \frac{\beta_{1}+\beta_{3}}{\rho_{0}}J_{3} \\
\dot{J_{3}}+2(\alpha_{2}-\alpha_{1})\beta_{2} & = & 0 \\
\dot{\beta}_{1}+\omega_{3}\beta_{2} & = & 0 \\...
Hi, I have derived a matrix from a system of ODE, and the matrix looked pretty bad at first. Then recently, I tried the Gauss elimination, followed by the exponential application on the matrix (e^[A]) and after another Gauss elimination, it turned "down" to the Identity matrix. This is awfully...
Homework Statement
Hi,
So I had a pretty long question solving a Linear ODE but now I've gotten stuck at this stage where I can't seem to get it into the right form to carry out partial fraction expansion
Homework EquationsThe Attempt at a Solution
[/B]
I'm quite sure that I what I have at...
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...
I need a translation of "Differentialgleichungen : Losungsmethoden und Losugen", I guess it is written in German. This book was referenced in Shepley L. Ross' book on ODE.
If the English translation is not unavailable, I am fine with a book that contains a "list" of special differential...
This past semester, I just took an introductory course on G.R., which translates to a lot of differential geometry and then concluding with Schwarzschild's solution. We really didn't do any cosmology. However, one of the themes that kept creeping up again and again is that in 4-dimensions...
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...
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 →...
I am trying to plot the trajectory of an asteroid in MATLAB using ode23. The only bodies in the system are The Sun, Earth, Mars and Jupiter and their orbital data has been loaded from data files. I have picked arbitrary initial conditions for the asteroid and believe my forces are correct. My...
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...
Homework Statement
I am trying to solve a system of 2 ordinary differential equations using matlab. However, I am not able to get numerical solutions from the code despite having keyed in all possible solutions.
Homework Equations
The equations I am given are:
dx/dt=A(x/t)+By...
Homework Statement
Liquid nitrogen is in a dewar connected to a vacuum pump. Initial pressure in a dewar is 1atm and saturated with gaseous nitrogen. If the vacuum pump started, it removes gas in it and the pressure in a dewar will be reduced under the saturation pressure of the liquid...
Hi there. I want to evolve a system of non linear coupled ODEs
\frac{dx}{dt} = \frac{-k}{x^5(56-y^8)^{9/2}}(85+y^{5} + y^{6})
\frac{dy}{dt} = \frac{-k}{x^4(56-y^5)^{7/2}}(44+y^2)
Let's say I have the initial conditions. What numerical method someone could use to solve this? adaptive step...
Given a linear homogeneous 2nd order ODE of the form $$y''(x)+p(x)y'(x)+q(x)=0$$ the general solution is of the form $$y(x)=c_{1}y_{1}(x)+c_{2}y_{1}(x)$$ where ##c_{1},c_{2}## are arbitrary constants and ##y_{1}(x), y_{2}(x)## are linearly independent basis solutions.
How does one prove that...
Homework Statement
\frac{dy}{dx}=y^2-1
y(0)=3
Homework Equations
\frac{dy}{dx}=f(y) \leftrightarrow \frac{dx}{dy}=\frac{1}{f(y)}
The Attempt at a Solution
\frac{dx}{dy}=\frac{1}{y^2-1}
dx=\frac{dy}{y^2-1}
\int dx=\int \frac{dy}{y^2-1}+C
x=\int \frac{dy}{y^2-1}+C
How do I integrate \int...
Homework Statement
imgur link: http://i.imgur.com/pb14Q4Q.png
Homework EquationsThe Attempt at a Solution
[/B]
The thing I don't understand is where the first two terms of each 2nd order ODE came about.
I understand that they are there because the coordinate system is rotating, but when...
Why these ODEs when applied some boundary conditions, like x = 0, their solution of the form Ax^k + Bx^(-k), B WILL have to go to zero?Like some problems which involve spherical harmonics...
Hi, I have the ODE y'''' - 3y' + 2y = 0
The characteristic equation is then $r^4 - 3r + 2 = 0$
So my 1st question, is there some easier way of finding the roots than long division?
I looked at the first and last terms to guess the roots (if real) might come from (r-1) , (r+1), (r+2), (r-2)...
Homework Statement
We have a driven pendulum described by the following differential equation:
\frac{d^2\theta}{dt^2} = \frac{-g}{l}\sin(\theta) + C\cos(\theta)\sin(\Omega t)
I need to turn this second order differential equation into a system of first order differential equations (then...
I have constructed a system of an arbitrary number of ordinary differential equations that describes my model at steady-state. There are ($i+1$) ODEs, with $i$ arbitrary.
Goal: I want to solve the resulting algebraic system (all equations set to $0$) and obtain an analytical expression that...
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...
Homework Statement
Sampled Data system
Using the forward Euler integration algorithm, convert these differential equations to a set of difference equations. Use a stept size of deltaT = 0.1s.
Homework Equations
x1(dot) = x2
x2(dot) = x3
x3(dot) = -2x1-3x2-4x3
y = 7x1-5x2
The Attempt at a...
I have an extremely messy system of differential equations. Can anyone offer any ideas for a general solution?
p(t) is a function of t, and A is a constant.
Hi Everyone,
I had a quick question. If you have an IVP ODE and you solve for the general solution first and you had fractions in it, could you multiply by a number to make it "easier" (whole number, rather than involving fractions) without violating the initial conditions?
Thanks
Hi there,
I have been using Leonard Susskind's lectures on classical mechanics to learn about Lagrangians and Hamiltonians, and decided to try to create a Lagrangian for the double pendulum and another pendulum-related system. I found the equations of motion, but they were unlike any...
Homework Statement
I stumbled upon a problem and i can't establish the ODE to solve it, from there on i believe i can solve the ODEs if they have regular analytical solving methods (translated from Spanish, will sound a bit weird)
Car race, 2 pilots (a and b) participate in a drag race. They...
Given a nonlinear system of eight autonomous differential equations with all variables and parameters living in the positive octant of real numbers:
$$dX_1/dt = \ldots\\
dX_2/dt = \ldots \\
\ldots \\
dX_8/dt = \ldots$$
and given that $\lim\limits_{t \to \infty} K(t) \to 0$ for $K(t) = X_3(t)...
Hello :)I have a system which consists of two coupled ODEs for which I want to solve.F'' *(1/b²) - α²*F = Ra*(1/b³)*G
G'' *(1/b²) - α*²G + Ra*(1/b²)*G = F'(1/b)
In these two equations F(z) and G(z) both depend on z. b is a constant, Ra is the rayleigh number which I need to keep as Ra in my...
Hello! (Wave)
I am looking at the following exercise:
Consider the initial value problem
$\left\{\begin{matrix}
x''(t)=x(t)\\
x(0)=a\\
x'(0)=b
\end{matrix}\right.$
Write it as a system of First-Order ODES with suitable initial values and show that Euler method can get unstable for a great...
Homework Statement
y''''''+y'''=t
Homework EquationsThe Attempt at a Solution
I got all the roots and solved the homo eq.
Then I tried to guess the partial eq and got At+B
However, I don't know how to proceed because the 6th derivative or the 3rd would be 0.
Hi,
Apart from the Euler's method, is there any other method (with better efficiency) that can let us solve an Ordinary Differential Equation of the form \frac{dy}{dx}= f(x,y)?
I understand how to integrate this: ∫y2dy.
I don't understand how to integrate this:
di(t)/dt = i(t)p(t)
intergrate((di(t)/dt/i(t))*dt = p(t)dt) (see this image: http://i.imgur.com/OdKI309.png)
how do you perform the intergral on the left, seeing as as it not dt, but di(t)?
thanks
Homework Statement
I'm trying to plot the steady state concentration of yA vs. x, yB vs x and yu vs x using centered finite difference method.
Homework Equations
The Attempt at a Solution
τ represents the dimensionless time variable, so steady state would mean that the left hand side of...