What is Seperation of variables: Definition and 36 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.
Homework Statement
I am trying to solve the given wave equation using separation of variables,
u_{tt} - 4u_{xx} = 4 for 0 < x < 2 and t > 0
(BC) u(0,t) = 0 , u(2,t) = -2, for t>0
(IC) u(x,0)=x-x^2 , u_t(x,0)=0 for 0\leq x \leq2
Homework Equations
We are told we will need to use,
x =...
Homework Statement
Boundary conditions are i) V=0 when y=0 ii) V=0 when y=a iii) V=V0(y) when x=0 iv) V=0 when x app infinity.
I understand and follow this problem (separating vars and eliminated constants) until the potential
is found to be V(x,y) = Ce^(-kx)*sin(ky)
Condition ii...
Homework Statement
House: a room (see figure) has perfectly isolated walls, except the two windows
where a convective heat exchange takes place (with the same transfer coefficient).
Outside temperature in front of a sun-faced wall-sized panoramic window is T1,
while at the back it is...
Homework Statement
The dielectric cylinder is radius R and thickness d. Origin is at the center of the cylinder, which is oriented along the z-axis. It has polarization P=pz∧I need to calculate the potential V(0,0,h) at h>d/2.
Homework Equations
σb=P⋅n∧...
Homework Statement
The cylinder has a radius a and is perpendicular to the electric field, E(r)=E(x_hat). It also carries charge Q. The potential is of the form V(r,φ)=A0+A0'ln(r)+∑(n=1 to ∞)((Ancos(nφ)+Bnsin(nφ))rn+(An'cos(nφ)+Bn'sin(nφ))r-n)
Homework Equations
V=-∫E⋅dl
The Attempt at a...
Homework Statement
I'm having issues with a Laplace problem. actually, I have two different boundary problems which I don't know how to solve analytically.
I couldn't find anything on this situations and if anybody could point me in the right direction it would be fantastic.
It's just Laplace's...
How do we know that separable solutions of Schrodinger equation (in 3d) form a complete basis? I understand that the SE is a linear PDE and therefore every linear combination of the separable solutions will also be a solution , but how do we know that the converse, i.e 'every solution can be...
Homework Statement
A uniform rod of length l has an initial (at time t = 0) temperature distribution given by u(x, 0) = sin(\frac{πx}{l}), 0 \leq x \leq l.
The temperature u(x, t) satisfies the classical one-dimensional diffusion equation, ut = kuxx
The ends of the rod are...
Homework Statement
Solve the given differential equation subject to the indicated initial condition.
(e^-y + 1)sinxdx=(1+cosx)d, y(0)=0Homework Equations
Basically we have to use separation of varaibles to solve before using initial value condition.The Attempt at a Solution
After separation...
I have two more loosely based questions about PDEs and the separation of variables technique:
In the intro of this chapter the author imposed that we "assume" the the solution to a set of special PDEs is:
U(x,t) = X(x)T(t) where X and T are the eigenfunctions. My question is how did...
Help -- Seperation of variables problem, multiple solutions.
Homework Statement
Suppose that dy/dx = √y and y(0) = 0. What is y(x)? There is more than one answer to this problem. You must list five correct solutions.
Homework Equations
Seperation of Variables/ integration
The...
Hi,
In Griffiths QM, it is claimed that to solve the Schrodinger Equation, we take the solution wavefunction \Psi(x,t) to be of a seperable form \psi(x)\phi(t).
He then says that a superposition of these seperable forms can always give us the general solution. Can someone help me prove that...
The method of separation of variables is used to solve the problem of partial diffrential equation. For example, when the partial differential equation is:
\frac{\partial u}{\partial t}-\alpha\frac{\partial^{2}u}{\partial x^{2}}=0
We could suppose that u(x,t) is a solution concerning...
Homework Statement
http://img18.imageshack.us/img18/8970/bose.png
The Attempt at a Solution
I'm on part b) where it asks which separation constation gives a harmonic time dependence. From part a) I deduced the equation \frac{d^{2}T}{dt^{2}}\frac{1}{T} = a constant. I'm choosing the constant...
Can anyone help me please or point me in the right direction, I am needing to find an exact solution for this equation by using separation of variables and compare them to answers i have calculated for Euler's method & Euler-Cauchy method. The equation is dx/dt=x^2/(t+1) when x(0)=1 and t=time...
Hello everyone how yall durrin!
Solve the following DEs by Seperation of Variables
eliminate natural logarithms and leave your final answer in implicit form
(3x+8)(y^2 +4)dx - 4y(x^2+5x+6)dy=0
by separation of variables i get
(3x+8)dx/(x^2+5x+6)=4ydy/(y^2 +4)
now I am having trouble...
Homework Statement
dy/dx=-ylny/x initial conditions y(x=1)=1
express answer in the form f(x,y)=0
Homework Equations
The Attempt at a Solution
i let u=ln y
du/dy=1/y
y*du=dy
subbing into equation gives int(y/yu du)=-int dx
which is equal to int(1/u du)=-int(1/x) dx...
Hey guys,
I was wondering about problem 12C.1 in Transport phenomena by Bird, Stewart and lightfoot.
The problem states that a block of material initially at uniform T0 is suddenely exposed to T1 at all surfaces.
Assume a solution of T=X(x,t)Y(y,t)Z(z,t)
any help with...
Suppose one is to find the stationary states of a particle in an infinite cubic well. Inside the box the time independent SE is:
- \frac{\hbar}{2m} \big( \frac{\partial ^2 \psi}{\partial x ^2 } + \frac{\partial ^2 \psi}{\partial z ^2 } + \frac{\partial ^2 \psi}{\partial z ^2 } \big)= E\psi...
I have some confusion about when separation variables can be applied to a PDE. Can it be applied on any PDE that can separated for any domain? If so, is the use of more "powerful" techniques simply used to save effort? (As you might have to use superposition several times!)
Newton's Law of Cooling (diff eq. -- separation of variables)
Homework Statement
Fresh coffee sitting in a room cooling...you know the routine.
Anyhow T(0) = 90degreesCelcius.
Room temp=25degrees Celcius
find k.
Then he asks us to use Euler's method to estimate coffee temp after five...
hey guys first post so sorry if this has already been asked :S
what exactly is meant by the separation of variables in the schrodinger equation? also what co-ordinate system would i use to solve for an electron in a hydrogen-like ion?
thanks
Hej,
This question is in the context of General Relativity problem. I'm attemping to compute the Killing Vectors for a Torus. After some juggling around I ended up with the following differential equation
\frac{d}{d \theta} \left( \frac{ (a+b \cos \theta) \sin \theta }{b} F(\phi) + g(\theta)...
Homework Statement
Homework Equations
After simplification, the PDE is
(b^2/a^2)(d^2 v/ d x^2) + (d^2 v/ d y^2) = -1
The Attempt at a Solution
Obviously, it can't be solved by separation of variables. And I also failed in similarity solution.
Homework Statement
Solve the following equation by a power series and also by separation of variables. Check that the two agree.
Homework Equations
N/A
The Attempt at a Solution
Power Series:
(1+x) \frac{dy}{dx} = y
(1+x) \frac{1}{dx} = y \frac{1}{dy}
The power...
Homework Statement
\frac{dV}{dt} = 20 - kV
By solving this show that
V = A + Be^{-kt}
Homework Equations
Well I am guessing there is a ln coming into play somewhere during the intergration
if the diff the bottom = the top then you get a ln(bottom)
The Attempt at a Solution...
Homework Statement
Create a 3D parabolic equation and solve it with the method of separation of variables.
Homework Equations
None
The Attempt at a Solution
\frac{\partial T}{\partial t}=\alpha [\frac{\partial^{2} T}{\partial x^{2}}+\frac{\partial^{2} T}{\partial...
Homework Statement
show the solution to dy/dx = [1+(1/y)]^1/2 is given by ∫[y^1/2]/[1+y]^1/2 dy = x + c
The Attempt at a Solution
I know this will be solved by separation of variables...
If i take the whole rhs over to the lhs I get...
dy/[1+(1/y)]^1/2 = dx
RHS no problems when...
Homework Statement
(x-1)y'=6y
Could someone please help explain to me how to do this problem.
Here's my attempt...
(x-1)y'=6y
(x-1)dy=6y
(x-1)=6y/dy
\int (x-1)dx=\int 6y/dy
((x^2,2)-x)= 3y^2+C
Then I try to get it to equal y and it comes out nothing like the answer. Am I even doing these...
[SOLVED] Seperation of variables in the 2 dimensional wave equation
I'd like to apologize right away for the terrible formatting. I was trying to make it pretty and easy to read but I guess I'm just not used the system yet and I had one problem after another. As you'll see at one point the...
[SOLVED] Seperation of variables - first order PDE
Homework Statement
I have the expression X'(x)/X(x) = cx. How do I separate the variables? It's the fraction on the left side that annoys me.
I know that X'(x) = d(X(x))/dx, but I can't use this here?
EDIT: Sorry for the mis-spelled title...
when is the separation of variables technique for partial differential equations valid? it seems to give a particular general solution (such as a general Fourier series, or series of legendre polynomials) to a problem depending which coordinate system that you are in?
Seperation of Variables (double check please :)
I have a final coming up, and I want to make sure I have this method down.
Q: For the second-order wave equation u_{tt}=u_{xx}, the substitution of u=A(x)B(t) will give second-order equations for A nd B when the x and t variables are...
This is the the first time I've encountered separation with partial differential equations. There are no worked examples, so I need some help to work through this problem. The question seems to be somewhat hand holding, since it seems to be THE introduction.
Q: Apply separation of variables...
i found the DE for a problem, and it was y'(t) = .06 - y/1040
and then the problem gave me the following hint:
Find the amount of sugar after t minutes. Note: When you solve by separation of variables, keep the coefficient of y on the right side and bring over to the denominator on the...
I've got a few 1st order ODEs which I have problems solving. I am new to the subject and self-taught so I may have a little difficulty absorbing. The question is...
1. \frac {dy}{dx} = \frac {y^3}{x^2}
for 1. I put it in the form,
x^2 dy = y^3 dx
\frac {dy}{y^3} = \frac...