Separation of variables Definition and 163 Threads

\frac{\partial z}{\partial t}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y} \mbox{Let }z=T(t)X(x)Y(y) T'(t)X(x)Y(y)=T(t)X'(x)Y(y)+T(t)X(x)Y'(y) \Rightarrow \frac{T'(t)}{T(t)}=\frac{X'(x)}{X(x)}+\frac{Y'(y)}{Y(y)} \Rightarrow \frac{T'(t)}{T(t)}=A \wedge...

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...

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 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...

Homework Statement Solve the given differential equation by separation of variables. Homework Equations \frac{dy}{dx} = \frac{xy + 2y - x - 2}{xy - 3y + x - 3} The Attempt at a Solution \frac{dy}{dx} = \frac{xy + 2y - x - 2}{xy - 3y + x - 3} = \frac{(x + 2)(y - 1)}{(x - 3)(y + 1)} (x -...

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...

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...

Hello, The following equation: \frac{\partial^2 u}{\partial r^2}+\frac{1}{r} \cdot \frac{\partial u}{\partial r}+ \frac{\partial^2 u}{\partial z^2} = 0 is solved by separation of variables assuming a solution of the form: u=R(r)Z(z) In other cases the assumed solution is of the...

Homework Statement Find an implicit and explicit solution for the given initial-value problem \frac{dx}{dt}=4(x^2+1) for x(\frac{\pi}{4})=1 \frac{dx}{dt}=4(x^2+1) \Rightarrow \frac{dx}{x^2+1}=4dt \Rightarrow \tan^{-1}x+C=4t Now I am a little stuck. Usually I just plug in my...

My book has a problem that requires you to separate variables (one side has all the y terms and one side has all of the x terms): \sin{xy'}=\cosx Equation after separation of variables: dy=\cot{x}dx My question is, how do you know that the y' is contained within the sine function or...

I do not understand the process of separating variables such as in derivatives. I thought that dy/dx is just the rate of change of y with respect to the independent variable x. Why can you physically move dx (like multiply it on both sides)?? What would "dy" be reffered to as then? Simply the...

u(r, θ) satisfies Laplace's equation inside a 90º sector of a circular annulus with a < r < b ; 0 < θ < π/2 . Use separation of variables to find the solution that satisfies the boundary conditions u(r, 0) = 0 u(r, π/2) = f(r) ; a < r < b u(a, θ) = 0 u(b, θ) = 0 ; 0 < θ < π/2 Consider all...

Homework Statement The temp. as a function of time of a metal rod obeys the following diff. eq. \alpha^2 \frac{\partial^2u(x,t)}{\partial x^2} = \frac{\partial u(x,t)}{\partial t} Use separation of variables to find u(x,t) in a rod of length 1 subject to the conditions u(0,t) = 0 ...

Homework Statement The edges of a square sheet of thermally conducting material are at x=0, x=L, y= -L/2 and y=L/2 The temperature of these edges are controlled to be: T = T0 at x = 0 and x = L T = T0 + T1sin(pi*x/L) at y = -L/2 and y = L/2 where T0 and T1 are constants...

Homework Statement if \nabla^2u = 0 in 0 \leq x \leq \pi, 0\leq y \leq \pi, boundary conditions u(0,y)=0, u(\pi,y)=cos^2y, u_y(x,0) = u_y(x,\pi)=0 Homework Equations I am required to show that u(x,y) = \frac{x}{2\pi} + \frac{cos2ysinh2x}{2sinh2\pi} The Attempt at a...