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 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
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...
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...
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...
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...
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) =...
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...
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...
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...
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...
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-...
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...
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
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...
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 →...
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'...
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...
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...
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...
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...
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...
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...
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
$$...
Homework Statement
\frac{dy}{dx}\:+\:ycosx\:=\:5cosx
I get two solutions for y however only one of them is correct according to my online homework
(see attempt at solution)
Homework Equations
y(0) = 7 is initial condition
The Attempt at a Solution
\int \:\frac{1}{5-y}dy\:=\:\int...
http://imgur.com/a/8QjoW
http://imgur.com/a/8QjoW
Hello-
I am trying to determine the dynamics of this linearly-damped hinge. Assuming that:
v(0) = 0
damping constant = b
door has mass = m
I was able to determine that:
∑Fx = -Fd * cos(45-θ/2) + Rx = m*dvx/dt
ΣFy = -Fd * sin(45-θ/2) - Fg +...
Hi
I am supposed to find solution of $$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} ...$$
$$...
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...
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.
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...
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...
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...
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...
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##...
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} =...
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...
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...
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...
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...
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...
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)
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...
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...
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...
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...
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...