# 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.

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1. ### I When does the separation of variables work

When studying the hydrogen atom, given that the potential depends only on the distance and not an any angle, we can do a separation of variables of the wavefunction as the product between a function depending only on the distance between particles (protons and electrons) and a spherical...
2. ### I Determining equilibrium solutions to differential equations

I am going through this, I noted that, i shall have a separation of variables, that leads to $$\left[\int \dfrac{1}{y(y-1)} dy\right]= \int \dfrac {1}{6} dt$$ and using partial fraction, i then have, $$\left[\int -\dfrac{1}{y} dy - \int \dfrac{1}{y-1} dy\right] = \int \dfrac {1}{6} dt$$...
3. ### I Heat Equation: Solve with Non-Homogeneous Boundary Conditions

Imagine you have a plane wall with constant thermal conductivity, that is the intermediate between two thermal reservoirs: The reservoir on the left is being kept at temp ##T_s##, and it is a fluid that has very high convective coefficient ##h##. As a result, the boundary condition at the...
4. ### Heat equation with non homogeneous BCs

I did a change of variable $$\theta(r,z) = T(r,z)-T_{\infty}$$ which resulted in $$\frac{1}{r}\frac{\partial }{\partial r}(r\frac{\partial \theta}{\partial r})+\frac{\partial^2 \theta}{\partial z^2}=0$$ $$\left.-k\frac{\partial \theta}{\partial r}\right\rvert_{r=R}=h\theta$$...
5. ### A Separation of variables is possible only in 11 coordinate systems?

I vaguely (strong word there because I can no longer remember the source, but the idea sticks in my head for 30 years now) recall reading (somewhere long forgotten) that method of separation of variables is possible in only 11 coordinate systems. I list them below: 1.Cartesian coordinates...

27. ### Wave equation, separation of variables and the Laplace transform

Homework Statement Homework Equations If i solve the wave equation using separation of variable and laplace tranform. Will i get the same answer ? The Attempt at a Solution
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### E&M separation of variables and Fourier

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...
29. S

### MATLAB Separation of variables in MATLAB

Hello, I haven't found any program that can be used to perform separation of variables on difficult PDEs. Is there such a method somewhere?
30. ### I Separation of variables for nonhomogeneous differential equation

Hi. I was wondering if it is possible to apply separation of variables for a function of space and time obeying a non homogeneous differential equation. In particular, the heat equation: ##\displaystyle \frac{\partial \Phi(\mathbf{r},t)}{\partial t}-\nabla \cdot \left [ \kappa(\mathbf{r})...
31. ### Question about separation of variables

I've solving some separation of variables exercises, and I have a doubt when it comes to the Laplacian $$u_{xx} +u_{yy} =0$$ I usually have a rectangle as boundary conditions, so I use the principe of superposition and arrive to $$\dfrac{X''(x)}{X(x)} = - \dfrac{Y''(y)}{Y(y)} = - \lambda$$...
32. ### MHB Method of separation of variables

Hello! (Wave) I want to check if the method of separation of variables can be used for the replacement of the following given partial differential equations from a pair of ordinary differential equations. If so, I want to find the equations.- $[p(x) u_x]_x-r(x) u_{tt}=0$ - \$u_{xx}+(x+y)...
33. ### B Separation of variables - rocket equation

hello there Im trying to do a derivation of tsiolkovsky's rocket equation, but i got stuck at the step when i have to use separation of variables (marked with red in the pic), i used maple to solve it, so i could get on with it, but i want to understand what is happening to solve this, so can...
34. ### Showing that a wavefunction can be written as a product

Homework Statement Let us look at a 3-dimensional potential box. Show, that the wave function in this situation can be written as the product of 3 single-argument functions. Homework Equations The 3D Schrödinger equation: -\frac{\hbar^2}{2m} \left( \frac{\partial^2...
35. ### Arbitrary constant in denominator

Homework Statement Find the general solution to the differential equation: Homework Equations Separation of variables for solving 1st order separable differential equation. The Attempt at a Solution Using separation of variables, I can write: My questions are: 1) Am I correct to...
36. ### I Using Separation of Variables to Modify Neutron Density Diff

I was overlooking page 47 of "The Physics of the Manhattan Project" 2.2 Critical Mass: Diffusion Theory, and author Bruce Cameron Reed reported that: Can anyone explain how Bruce Cameron Reed got from (2.18) to (2.19) I tried plugging ## N(r,t) = N(r) N(t) ## into (2.18) to get (2.19), but it...
37. ### PDE — lost on this separation of variables problem

<< Mentor Note -- thread moved from the technical math forums >> I am getting stuck on this partial differential equation. Ut = Uxx - U + x ; 0<x<1 U(0,t) = 0 U(1,t) = 1 U(x,0) = 0 Here is my work so far : U = e-tw + x gives the new eq wt=wxx to get rid of boundary conditions : w=x+W Wt=Wxx...
38. ### I Basic Separation of Variables problem

I'm self-teaching through Tenenbaum & Pollard's "Ordinary Differential Equations", and for some reason I'm completely stuck on one of the problems, Ch.2, lesson 6, problem #6: Find a 1-parameter family of solutions for [...] the differential equation: 6) yx2dy-y3dx = 2x2dy. I didn't have...
39. ### Solving for electric potential using separation of variables

Homework Statement Given two two grounded concentric spherical shells with radii a,b (a<b) and a point charge q between them at a<r=R<b find: 1.The surface charge density of the point charge using the delta function, assume the charge is on the z axis 2.By using the separation of variables...
40. ### Solve PDE by separation of variables

Homework Statement Solve ∇2T(x, y) = 0 with boundary conditions T(0, y) = T(L, y) = T0 T(x, L/2) = T(x, -L/2) = T0 + T1sin(πx/L) Homework EquationsThe Attempt at a Solution Set T(x, y) = X(x)Y(y) Then ∇2T(x,y) = (∂2X/∂x2) Y + (∂2Y/∂y2) X = 0 Rearrange to find two separate ODEs: d2X/dx2 =...
41. ### Solve PDE with separation of variables

Homework Statement The wave equation for ψ(t, x) in 3D is ##\frac{\partial ^2 \psi}{\partial t^2}## - Δ ##\psi =0## Let ϒ(x) satisfy Δϒ = λϒ where λ<0. The x is in bold presumably to indicate it is in 3D, so represents also y and z? Show there is a solution of the form ψ(t, x) = sin(ωt)ϒ(x)...
42. ### How to Separate the Wave Equation into Three ODEs Using Separation of Variables?

Homework Statement The wave equation for ψ=ψ(t,x,y) is given by ##\frac{\partial ^2 \phi}{\partial t^2} - \frac{\partial ^2 \phi}{\partial x^2} - \frac{\partial ^2 \phi}{\partial y^2}## Use separation of variables to separate the equation into 3 ODEs for T, X and Y. Use the separation...
43. ### Can All Differential Equations Be Separated?

I just wanted to check something. The equation ∂2φ / ∂x2 + ∂2φ / ∂y2 = sin(xy) Was given as an example of a separable equation. I can't separate it, and I found online that to use separation of variables the equation should be linear, which this isn't? Is there a way of separating this?
44. ### Separation of variables and potential

Homework Statement A potential satisfies ##\nabla^2 Φ = 0## in the 2d slab ## -\inf < x < \inf ##, ##-b < y < b ##, with boundary conditions ## Φ(x, +b) = +V_s(x)## on the top and ##Φ(x, b) = -V_s(x)## on the bottom, where[/B] ##V_s (x)= -V_0 for -a<x<0## ##V_s (x)=+V_0 for 0<x<a## (a) what...
45. ### Separation of Variables Problem

Homework Statement A square is made up of four plates with a potential of zero on the top and bottom plates at (x,L/2) and (x,-L/2), and a potential of cos(πy/L)+cos(3πy/L). Find the potential and electric fields inside the square. The Attempt at a Solution I start with...
46. ### Another Diff Eq's one: am I wrong?

Homework Statement Sorry to be bringing these in quick succession, but this one really has me perplexed. Is it possible that both the solution manual and I have two different but valid answers? I don't want to just go assuming that I'm right...but I think by subsuming certain parts of the...
47. ### Explaining the Solution for Separation of Variables PDE with Initial Condition

Homework Statement Homework EquationsThe Attempt at a Solution I managed to do the first part of the question. But I'm not sure how to find u(x,t) with that initial condition. The solution says; "since ##u(x,0) = \sum_{n=1}^\infty a_{n}\sin{(n\lambda x)}## Then it follows by linearity that...
48. ### An equation for the path that the shark will swim on

Homework Statement [/B] A shark will in the direction of the most rapidly increasing concentration of blood in water. Suppose a shark is at a point x_0,y_0 when it first detects blood in the water. Find an equation for the path that the shark will follow by setting up and solving a...
49. ### Separation of Variables in PDEs

I am using the text by Farlow to study elementary methods of solving PDEs, and there is a point in his illustration of separation of variables where I am not seeing something. I am clear on everything that comes after and before this point, but after having returned to a certain step a few times...
50. ### Separation of variables to solve Schrodinger equation

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...