Separation of variables Definition and 163 Threads

Homework Statement So I'm doing a question from one of my past exams as attached, there are no copy right issues with this document that I know of and have asked my lecturer who wrote the exam and he said I am welcome to upload it. The question is 1)b)iv), my attempt is attached. I end up with...

If $$u=\frac{1}{2} E^2$$ and $$v=\frac{1}{2}B^2$$ and we have that $$\frac{\partial L}{\partial u} \frac{\partial L}{\partial v} = -1$$ The author says: to obtain explicit solution of the above, one must resort to techniques such as separation of variables in particular coordinate systems. For...

So I'm currently taking electricity and magnetism and I'm expected to know how to perform a separation of variables on laplace equation in 2 dimensions.I have taken Zero differntial equations courses and I literally have no freaking idea what's going on. The book I use doesn't spend any time...

My friends and I are in our first senior-level physics course at the University of Alabama in Huntsville, Introductory E&M. At the moment, we're working on using separation of variables to calculate electric potentials inside different objects given certain boundary conditions. One, however, is...

$$(2xy-3y)dx-({x}^{2}-x)dy=0$$ ans. $$xy(x-3)=C$$ ty

I am curious how legitimate a solution Separation of Variables tends to give. I've been working problems out of Griffith's book on Electromagnetism, and am often uneasy as to the way things are done. I have two specific issues. The first, is that in spherical it is often necessary to remove...

I'm having some difficulty setting up a problem. I'm trying to model the temperature of a thermistor connected to a constant current source. The thermistor's resistance varies with temperature, so with a fixed current, I would expect to see the thermistor's temperature to oscillate with time...

Hey guys, Can anyone help me out with these questions? The first one has a positive initial value. Separation of variables and integrating gave me: |y+3| = k√[(t^2) + 1)] Ultimately, I got k= √5 and thus y=√5√[(t^2) + 1)] - 3. Also, for the second one, I used a similar process, found...

Why is it assumed that the method of separation of variables works when the boundary conditions of some boundary valued problem are homogeneous? What is the reasoning behind it?

Hey! :o I have a question.. (Wasntme) When we have the following boundary value problem: $$u_{xx}+u_{yy}=0, 0<x<a, 0<y<b (1)$$ $$u_x(0,y)=u_x(a,y)=0, 0<y<b$$ $$u(x,0)=x, u_y(x,b)=0, 0<x<a$$using the method of separation of variables, the solution would be of the form $u(x,y)=X(x) \cdot Y(y)$...

Can someone please help me solve the following using separation of variables: dy/dx = (xy + 3x -y-3)/(xy -4x+6y-24) so that the solution is written in the form: ((x+6)/(y+3))^7 =

Can someone please help me to calculate the following using separation of variables: dy/dx = x*(1 - y^2)^(1/2) to that the solution is in the form: y =

I am studying Laplace's equation in my electrodynamics course (using griffiths intro to electrodynamics). I am watching a youtube video stepping through the separation of variables method for solving the PDE. It seems to be a common PDE that comes up repeatedly in physics (Helmholtz eqn, Poisson...

"Critiquing" separation of variables method for PDE. I am currently taking a course in PDE's and it has been very "applied" and not so much theory based. I can say its been separate this separate that separate this separate that… Enough! We are always "separating variables" and it always...

Homework Statement utt = uxx -(25/4)cos((5/2)x) ux(0,t) =1 u(pi,t)= pi u(x,0)=x ut(x,0)=0 Homework Equations u(x,t)=v(x) + w(x,t) The Attempt at a Solution This is what I did so far: u(x,t)=v(x) + w(x,t) u(x,0) = v(x) +w(x,0) when t is large: vxx - (25/4)cos((5/2)x) = 0 vx =...

Homework Statement Hey guys, I have this problem which I am having a hard time solving. $$u_{tt} -x^2u_{xx} = 0$$ $$1<x<2 \hspace{4mm} t>0$$ $$u(x,0)=0$$ $$u_t(x,0)=g(x)$$ $$u(1,t)=0=u(2,t)$$ Homework Equations $$u_{tt} -x^2u_{xx} = 0$$ $$1<x<2 \hspace{4mm} t>0$$...

Solve given differential equation by separation of variables \frac{dy}{dx}=\frac{xy+3x-y-3}{xy-2x+4y-8} So separate x and y terms (xy-2x+4y-8) dy = (xy+3x-y-3) ugh I'm stuck:(

Homework Statement Using the technique involving variable separation, solve the following differential equation and use the initial condition to find the particular solution \frac{dy}{dt} = \frac{1}{y^{2}} y(0) = 1 Homework Equations The Attempt at a Solution To be honest...

Solve the DE by using separation of variables \frac{dy}{dx} = e^{3x+2y} Break up e^{3x+2y} = e^{3x}e^{2y} Move x's and y's to their own side of the equation. \frac{1}{e^{2y}} dy = e^{3x} dx Integrate both sides of the equation to get \frac{-e^{2y}}{2x}=\frac{e^{3x}}{3}+C I don't know how to...

Solve the differential equation by separation of variables x \frac{dy}{dx} = 4y becomes \frac{1}{4y} dy = \frac{1}{x} dx Integrate to get \frac{1}{4} \ln{|y|} = \ln{|x|}+C I'm stuck here because I want to raise e to the power of both sides of the expression like e^{ \frac{1}{4} \ln{|y|}} =...

Homework Statement In the problem, we are to consider two candles, call them C1 and C2, with different heights and different thicknesses. Call the height of C1 H1, and for C2, call it H2. The taller candle burns can burn for 7/2 hours, and the short one, 5 hours. After two hours lapses, the...

Homework Statement dL/dp = L/2, L(0) = 100. Find the solution to the differential equation, subject to the given initial condition. My textbook says the answer is L = 100ep/2, but I don't know how to get that answer (or e for that matter). Homework Equations ? The Attempt at a...

I'm having troubles with PDE. Apply separation of variables, if possible, to found product solutions to the following differential equations. a) x\frac{\partial u}{\partial x}=y\frac{\partial u}{\partial y} I suppose that: u=X(x) \cdot Y(y) Then: xX'Y=yXY' xX'/X=yY'/Y So xX'/X=yY'/Y=c because...

I've found many articles online that explain how to solve the Schrodinger equation for a potential dependent on x, but not for one dependent on t. A couple articles said that you could not use separation of variables to solve the Schrodinger equation with a time dependent potential, but they did...

Why is the "Separation of Variables" method valid? Hey guys, Lately I have been focusing on some question that have annoyed me for some time. One of these questions is: Why is the method of separation of variables valid when solving some PDE? Usually smmetry arguments are presented, and...

Suppose you have some partiel DE describing a physical system with 2 degrees of freedom (e.g. the SE). If you try separation of variables you get something like: Hg(x)h(y) = Eg(x)h(y) now you can separate this to two equations, but the energy has to go in one of them. Is the final...

Homework Statement Find all solutions. Solve explicitly for y. y^{'}=y^{2}-y Homework Equations The Attempt at a Solution Case where y'=0 0=y(y-1) y=0,1 when y(t)=0 Case where y'\neq0 y'=y^{2}-y \frac{1}{y^{2}-y}y'=1 \int\frac{1}{y^{2}-y}y'dt=∫1dt \int\frac{1}{y^{2}-y}dy=t+c Cant...

Homework Statement Hopefully no one will mind me posting this as an image. But here it is in tex: Using separation of variables, find the function u(x,t), defined for 0\leq x\leq 4\pi and t\geq 0, which satisfies the following conditions: \frac{\partial^2 u}{\partial...

Homework Statement Solve the diffusion equation: u_{xx}-\alpha^2 u_{t}=0 With the boundary and initial conditions: u(0,t)=u_{0} u(L,t)=u_{L} u(x,0=\phi(x) The Attempt at a Solution I want to solve using separation of variables... I start by assuming a solution of the form...

Homework Statement I must find the oscillations of a circular membrane (drum-like). 1)With the boundary condition that the membrane is fixed at r=a. 2)That the membrane is free. Homework Equations The wave equation \frac{\partial ^2 u }{\partial t^2 } - c^2 \triangle u =0...

Homework Statement Homework Equations Assume the solution has a form of: The Attempt at a Solution It looks like a sine Fourier series except for the 2c5 term outside of the series, so I'm not sure how to go about solving for the coefficients c5 and c10. Any idea?

Homework Statement I'm solving a problem where a conducting pipe with a square cross section is being analyzed to find the potential everywhere in space. The pipe lays along the z-axis, so we're really concerned with the x-y plane. My issue isn't so much the general solution via separation...

Homework Statement V_o varies as V(t) = 2000e^(-100t) volts for t > 0 i1(0) = -6A i2(0) = 1A thus i0(0) = 5A Find i0(t) for t > 0. I found Leq = 4H V(t) = L di/dt I separated variables and integrated, got 5e^(-100t) but the software is telling me it's wrong... supposedly the answer is...

Hi all Quick one, if one had an equation y' = x on could simply separate the variables and integrate. Now it the equation y'' = x you would use separation of variables what drives this? Also y'' =0. Is the same as. y''dx =0 dx Why is this legal?Thanks in advance

"separation of variables", but for 2nd order Ok, I know how to separate variables in solving an ODE. I am unable to understand a solution I have for a problem which was the result of reduction of order- we end up with u''*sinx-2u'*cosx=0 so turn this into u''/u'=-2cosx/sinx At this point I...

Not really a specific problem, but just a general question: Does anyone have any good references (preferably online) for solving E&M problems with this method? I'm using Griffith's Electrodynamics book for my class and I'm trying to get ready for a final. This is the only part I'm having...

Homework Statement solve dy/dx=e^(3x+2y) by separation of variables The Attempt at a Solution \frac{dy}{dx}=e^{3x+2y} \frac{dy}{dx}=e^{3x}e^{2y} e^{-2y}dy=e^{3x}dx \int e^{-2y}dy=\int e^{3x}dx e^{-2y}=-\frac{2}{3}e^3x + C -2y = ln(-\frac{2}{3}e^{3x}+C)...

Homework Statement Use separation of variables to find the solution to Laplaces equation satisfying the boundary conditions u(x,0)=0 (0<x<2) u(x,1)=0 (0<x<2) u(0,y)=0 (0<y<1) u(2,y)= asin2πy(0<y<1) The Attempt at a Solution I am able to perform the separation of variables...

The normalization condition is: ∫|ψ|^{2}d^{3}r=1 In spherical coordinates: d^{3}r=r^{2}sinθdrdθd\phi Separating variables: ∫|ψ|^{2}r^{2}sinθdrdθd\phi=∫|R|^{2}r^{2}dr∫|Y|^{2}sinθdθd\phi=1 The next step is the part I don't understand. It says: ∫^{∞}_{0}|R|^{2}r^{2}dr=1 and...

Homework Statement dP/dt=P-P^2 Homework Equations The Attempt at a Solution I know you can separate this and after i did that and did my partial fractions i got t + C = ln(P) + ln(1-P) but i don't know what to do from here i figure you take the e of both sides at some point...

Homework Statement Solve the following equation by separation of the variables: y' tan-1x - y (1+x2)-1 = 0 Homework Equations The Attempt at a Solution I am not sure if tan-1x stands for arctan x or (tan x)-1. (This has been taken out a book.) Any help on this would be...

Solve the following first order, ordinary differential equations using separation of variables: dy/dx = y^2 x subject to y=-1 when x=0 the correct answer is: y = -2/x^2 + 2 i cannot seem to get this answer, after i separate the variables and integrate both sides i get: y^2 x^2/2...

Homework Statement Given the partial differential equation: ∂2u/∂x2 = ∂2u/∂t2 , where x[0;L] Use separation of variables to find the solution that satisfies the boundary conditions: ∂u/∂x (x=0) = ∂u/∂x (x=L) = 0 Homework Equations The separation of variables method. The...

Homework Statement Solve the differential equation: y' + cos(x)y = cos(x) The Attempt at a Solution y' = cos(x)(1 -y) \frac{dy}{1-y} = cos(x) dx -ln|1-y| = sin(x) + C \frac{1}{1-y} = e^{sin(x)} + C 1-y = \frac{1}{e^{sin(x)} + C} y = \frac{-1}{e^{sin(x)} + C} + 1 Have I done this...

Homework Statement use separation of variables to solve the differential equation x^2dy/dx=y-xy with the initial condition of y(-1)=-1 Homework Equations The Attempt at a Solution after i separated and integrated i got the answer y=e^(-1/x-lnx+c) the answer in the book is y=e^-(1+1/x)/x i...

\displaystyle\frac{dN}{dt} = rN\left(1-\frac{N}{K}\right)\displaystyle\int\frac{KdN}{N\left(K-N\right)} = \int rdt \displaystyle K\int\frac{dN}{N}-K\int\frac{dN}{K-N}=r\int dt Now, I obtain: K\ln\left(\frac{N}{K-N}\right) = rt+c \left(\frac{N}{K-N}\right)^K=C_0r^{rt} The final solution is...

Homework Statement Taking the equation of motion for a rocket launched from rest in a gravitational field g, m\dot{v} = -\dot{m}v_{ex} - mg , and knowing that the rocket ejects mass (fuel) at a constant rate \dot{m} = -k (where k is a positive constant), so that m = m_{o} - kt . Solve...

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.

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...