What is Separation of variables: Definition and 171 Discussions
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.
Dear Everyone,
I have a question about the separation of variables for non-central potentials (r, \theta, \phi). In spherical coordinates, such a potential V(r,\theta)=u(r)+f(\theta)/r^2 can be separated. Who knows it could also be separated in other coordinates? Many thanks.
Hi Everyone,
I am trying to solve the partial differential equation given below:
\Delta^2\phi(x,y,z)=\frac{qf(x,y,z)}{\epsilon}
where f(x,y,z)=1 at one point and zero elsewhere.
This is the poisons equation for a point charge inside a conducting box.
Can this be solved using the...
Homework Statement
Solve the following separable equation
\frac{dy}{dx} = \frac{y}{x(x-1)}
Homework Equations
\int\frac{1}{x}dx=ln(x)
The Attempt at a Solution
See attachment
Im getting y=(\frac{x-1}{x})^{c} as my answer when in fact the answer the tutor gave is...
Homework Statement
Sorry don't know how to use the partial symbol, bear with me
partial u wrt t=2*(2nd partial u wrt x)
Boundary conditions:
partial u wrt x (0,t)=partial u wrt x (1,t)=0
Initial conditions
u(x,0)=x(1-x)
Homework Equations
I get an answer that is different...
Im doing separation of variables now and I am stuck...ive come up to this stage
-5 ln 2-3i = t^2/2 + C
my problem is do i need to divide the C and t^2/2 by -5? or don't divide the C?
please help me..
thnks
a) Derive a formula for solutions of the ode equation
b) If (t0, x0) lies in the subset of the (t,x)-plane given by x>1 write down a formula for the unique solution of the equation below through this point.
c)Give a formula for two solutions to equation below through the point (0,1) in the...
See attached image for the question and my working. Hopefully you can read it OK, I had to resize it to fit to the allowed dimensions.
I'm unsure how to proceed or if I have done something wrong previously - the initial and boundary conditions are tripping me up. The boundary conditions in red...
Just quick question about sep of variables..
say have function U(x,y)=X(x)Y(y)
when do separation of variables end up with some generic case that looks like:
X''/X=Y'/Y=lamda
my question is (and I think I know now the answer but would like confirmation), is what sign should the lamda...
Homework Statement
Use separation of variables to find a general series solution of
u_t + 4tu = u_{xx} for 0 < x < 1, t> 0 and u(0,t) = u(1,t)=0.
Homework Equations
The Attempt at a Solution
Looking for a solution of the form u(x,t) = X(x)T(t) implies that \frac{T'}{kt} - \frac{X''}{X} = 0...
Homework Statement
As part of the solution to a HW problem of mine, I have to solve the PDE
p_t = -vk^2 p - k \delta p_k,
where p = p(k,t) and v,\delta are known constants.
Homework Equations
I tried to look for a solution of the form p(k,t) = K(k)T(t) and found one, but I'm not sure if I...
Homework Statement
In the midst of Forced Vibrating Membranes and Resonance Utt = c^2*delsquared(U) + Q(heat source)
Arrive at eigenfunction series solution where the coefficients are given by
d^2/dt^2 (A_n) + c^2*lambda_n*A_n = q_n
Homework Equations
according to the book, I am supposed to...
Homework Statement
Solve the boundary value problem for a string of unit length, subject to the given conditions.
f(x)=0.05sin \pi x, g(x)=0, c=\frac{1}{\pi}
Homework Equations
Model: u(x,t)=X(x)T(t)
Which yields two separated equations by the one dimensional wave equation.
X''-kX=0 and...
Given the Cauchy problem
\begin{cases}
u_{xx}+u_{yy}=0, & 0<y<\infty, x\in\mathbb{R} \\
u(x,0)=0 & \mbox{on }\{ x_2=0\}\\
\frac{\partial u}{\partial y}=\frac{1}{n}\sin (n y) & \mbox{on }\{ x_2=0\}\\
\end{cases}
I am given that, by using using separation of variables, the solution ought...
Homework Statement
I'm supposed to find a nontrivial solution to tx'' + (t-2)x' + x = 0, x(0) = 0. You don't really need to know that but just in case.
I got to this point:
(s+1)X'(s) + 4X(s) = 0
Now I need to separate variables to find a solution but I've been working on this for two...
dy/dx = 3[(y-1)^(1/3)]
with initial values: y(0)=1
ultimately I end up with y=sqrt((2x)^3) + 1 or ((2x)^3/2) + 1
book answer: y= 1+((3x)^3/2)
steps: separation of variables...
((y-1)^-1/3)dy = 3dx
after u-subs, where u = y-1...
3/2((y-1)^2/3) = 3x + C...
Homework Statement
A cubical box (sides of length a) consists of five metal plates, which are welded together and grounded (Fig 3.23). The top is made of a separate sheet of metal, insulated from the others, and held at a constant potential V0. Find the potential inside the box...
Homework Statement
We have a 2-dimensional box with only one side at a potential V0. The other 3 sides are grounded. The box is a square with top and bottom at y=a/2 and –a/2 and sides at x=±a/2. Find V(x,y) (it should contain cos and sinh).
Homework Equations
The Attempt at a...
Hi. This is my first post in PF. I'm an undergraduate student of Electronics Engineering with strong interest in math & physics (and weak understanding of them :P).
One of the things I always hate is when in some book (or some lectures) ODEs or PDEs are solved after the magic words "[...] and...
Homework Statement
y' + (2/x)y = 3/x^2
Homework Equations
separation of variables
The Attempt at a Solution
First I turned it into
dy/dx + (2/x)y = 3/x^2 dx
then multiplied both sides by dx
dy + (2/x)y = 3/x^2 dx
I then tried to divide both sides by 2/x and got...
Homework Statement
du/dt=d2u/dx2, u(0,t)=0, u(pi,t)=0
u(x,0) = sin^2(x) 0<x<pi
Find the solution
Also find the solution to the initial condition:
du/dt u(x,0) = sin^2(x) 0<x<pi
The Attempt at a Solution
From separation of variables I obtain
u(x,t) = B.e^(-L^2t).sin(Lx)...
Hi. I believe I understand separation of variables for a first order DE. But can anyone tell me how to use it on a second order DE? In particular I have been looking at this example
http://en.wikipedia.org/wiki/Integrating_factor#General_use"
where it is claimed that one can use separation of...
Homework Statement
Find the general solution for this differential equation.
dy −2x^2 + y^2 + x
dx = x yHomework Equations
The Attempt at a Solution
dy/dx = (−2x^2 + y^2 + x) / (x y)
let y^2 = v
dy/dx = v + x dv/dx
v + x (dv/dx) = (-2x^2 + v^2 x^2 + x ) / v x^2
=> x (dv/dx)...
Homework Statement
The differential equation I have is dy/dx = (xy + 2y - x - 2)/(xy - 3y + x - 3). I need help getting started. Neither the top nor the bottom can be factored, so I don't know what to do next. Can anyone give me a push? All I know is that I need to use separation of variables.
1. Solve the wave equation u_(tt) = 4u_(xx) on the interval [0, π] subject to the
conditions
u(x, 0) = cos x, u_t(x, 0) = 1, u(0, t) = 0 = u(π, t).
Homework Equations
3. Hello. This appears to be a common separation of variables question. Only problem is after using...
When solving a pde using this method how do you know what ORDER to use the initial/boundary conditions given to you?
E.g. if you are asked to solve the wave equation given u(x,0), u'(x,0), u(0,t), u(l,t) how do you know what order to use these in (particularly the first two)
Homework Statement
Use separation of variables to solve (x+2y)y'=1 y(0)=2Homework Equations
u=2y+x >>I did not know how to start this, so i looked at the back of the book and it said to use that substitution
y=(u-x)/2, du=2dy+dx, dy=(du-dx)/2
The Attempt at a Solution
so i got the following...
a) Show that the Hamiltonian for the quantum harmonic oscillator in 3D is separable, b) calculate the energy levels.----a) If it's separable H = H_x + H_y + H_z, so do I just re-arrange the kinetic and potential terms of the Hamiltonian in this case? that seems kind of trivial, as if I'm...
Homework Statement
Solve the problem.
utt = uxx 0 < x < 1, t > 0
u(x,0) = x, ut(x,0) = x(1-x), u(0,t) = 0, u(1,t) = 1
Homework Equations
The Attempt at a Solution
Here is what I have so far but I'm not sure if I am on the right path or not.
u(x,t) = X(x)T(t)...
Please have mercy on my non-mathematical mind...
I am struggling so hard with Differential Equations.
This is my third time to take the class and I still feel like I am walking through the woods at midnight on an extremely dark night.
This is the problem that I am working on:
dy/dx =...
Homework Statement
This is a try for the solution of Laplace Equation. We have to calculate the potential distribution in a cylinder coordinate. However, there is a step really bring us trouble. Please go to the detail. You can either read it in the related URL, or in my PDF attachment...
this may seem like a simple question but how does one know that separation of variables for solving linear PDE's will work. What i mean is that it seems to pick out a form of the solution to a given problem (I have heard that linear PDE's have an infinite number of functions of a particular...
Homework Statement
Solve the heat flow problem using the method of separation of variables:
Homework Equations
PDE:\frac{\partial u}{\partial t}=k\frac{\partial^{2} u}{\partial t^{2}}
for 0<x<L, 0<t<\infty
BC's:\frac{\partial u}{\partial x}(0,t)=0,\frac{\partial u}{\partial x}(L,t)=0...
Hi all,
For my thesis I would like to solve the following second order nonlinear PDE for V(x,\sigma,t):
\frac{1}{2}\sigma^2\frac{\partial^2 V}{\partial x^2}+\frac{1}{2}B^2\frac{\partial^2 V}{\partial \sigma^2}+a\frac{\partial V}{\partial \sigma}=0,
subject to the following boundary...
Homework Statement
Hi, I don't really understand separation of variables very well, and I was hoping to do get my mind more clear on the following question:
(Q) Use separation of variables to find all the separable solutions of the equation:
d²y/dt² -c²(d²y/dx²) + w²y = 0
where 'w'...
So my book says that to solve a PDE by separation of variables, we check the three cases where λ, the separation constant, is equal to 0, -a^2, and a^2. But in this particular problem, instead of substituting λ=0, λ = a^2, λ= -a^2, they substitute the entire coefficient of X, (λ-1)/k =0, (λ-1)/k...
To solve a separable ODE like this I would simply multiply each side by dx and then integrate both sides. However, I know that it is only notational convenience that allows me to do this, and what's really going on is slightly more complicated.
Take this DE for example...
Good day,
I have to separate the variables of the formula (dy/dx) + 1 = - (y/x)
so I can determine the solution of the differential equation.
I get:
(dy/dx) + 1 = - (y/x)
(dy/dx) = - (y/x) - 1
(dy) = (- (y/x) - 1)dx
Though I cannot get rid of the y at the side of dx...
Hi, I am new to the forum. I've encountered a couple problems with separation of variables in cylindrical coordinates.
Problem #1: a clindrical surface of radius R is oriented along the z-axis, and is split into two
conducting half-cylinders. The potential satisfies the...
Electromagnetism just got weird. REALLY weird. Everything was going great until we hit this new chapter on separation of variables. I don't remember doing this kind of stuff in my DiffEqs class.
Frankly, I'm feeling overwhelmed. I have a midterm at the end of this week, and I feel as though...
"A sphere of homogeneous linear dielectric material is palcced in an otherwise uniform electric field E. Find the electric field inside the sphere."
Griffiths uses separation of variables to solve laplace's equation in the interior of the sphere. I have two questions.
(1) How can you try...
Homework Statement
Solve the 2-D time-independent Schrödinger equation with V (x,y) = 0:
Homework Equations
-ћ2/2m ( ∂2Ψ(x,y)/∂x2 + ∂2Ψ(x,y)/∂y2 ) = EΨ(x,y)
The Attempt at a Solution
I started by getting -ћ2/2m to one side:
( ∂2Ψ(x,y)/∂x2 + ∂2Ψ(x,y)/∂y2...
Homework Statement
so here's my equation:
dy/dx=(xy+3x-y-3)/(xy-2x+4y-8)
so what i did first was factor out the right side
=(x+1)(y-3)/(x+4)(y-2)
then i did a bunch of manipulation to get the ys on one side and the xs on another (i won't write this out right now but if anyone...
Homework Statement
Not sure if you guys can get this link
http://www.maths.uwa.edu.au/devsite/Units/math3341-s1-2008-crawley/assignments-solutions/Sheet%204
should be able to.
Question is question two.Homework Equations
Not many besides the general separation of solutions sort of thing but I...
Homework Statement
y'=xsec^2(x^2)
2. The attempt at a solution
dy/dx=xsec^2(x^2)
dy=xsec^2(x^2)dx
\intdy=\intxsec^2(x^2)dx
lny= (here i'll do a u substitution)
----
u=x^2 du=1/3x^3dx
... and here's my problem. It seems like that creates a very difficult u-sub to try and manage...
Hello !
I'm having a hard time finding the exact hypotheses which would allow me to use the separation of variables method for partial differential equations.
I want a clear statement telling me 'when you have that very kind of partial differential equation (with precise boundary and...
Homework Statement
Sorry for the mis-spelled title - it's "series".
Please take a look at
http://www-solar.mcs.st-and.ac.uk/~alan/MT2003/PDE/node21.html
In step 2.60, when he wants to find the coefficient B_n, the argument in the sine-function does not contain a "2". In my book, the...
Homework Statement
Please take a look at:
http://www-solar.mcs.st-and.ac.uk/~alan/MT2003/PDE/node21.html
Look at step 2.53. Can you explain to me how c^2*T'/T = k becomes T' = k*T*c^2?
The Attempt at a Solution
I don't get it. What am I missing here?
Homework Statement
Using separation of variables determine if the solution escapes to infinity in finite time or infinite time?
y'(t)=1+\frac{y(t)}{2}
y(0)=.5
Homework Equations
Knowing how to do separation of variables.The Attempt at a Solution
Here is my attempt, but I get stuck...