Separation of variables Definition and 163 Threads

For conventional PDEs like diffusion, waves, it seems the standard way to solving them is in two steps. 1. Use separation of variables method to make them into ODEs 2. Use eigenvalues and eigenfunctions theory on ODEs to construct the final solution consisting of an infinite number of...

"Use separation of variables to find particular solutions of u_t-u_{xx}-2u_x=0, 1<x<2, 0<t, u(1,t)=u(2,t)=0 hint: change coordinates" I can't find the solution. The equation seems already separated, so all I need to do is to find a change of variables, I think. But I can't find one that...

Let be the integral equation: g(s)g(p)g(u)= \int_{0}^{\infty}dx\int_{0}^{\infty}dy\int_{0}^{\infty}dzK(sx)K(py)K(uz)f(x,y,z) then my question is if we could "seek" for a solution in the form: f(x,y,z)=A(x)A(y)A(z) where the function A satsify (for x y and z) the integral...

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xdv/dx=(1-4v^2)/3v I used separation of variables to get x/dx=(1-4v^2)/3v dv I'm not sure if that's even right. But if it is right, how do I integrate that?

Hello, Could someone please help me to simplify my solution to my ODE? Here is the solution I get when I check it using Maple 10, http://img524.imageshack.us/img524/415/ode2hx.jpg Here are my steps: \left( {1 + x^3 } \right)\frac{{dy}}{{dx}} - 3x^2 y = 0 \left( {1 + x^3...

hi i have been trying to solve the diffusion equation using separation of variables. i know the answer should turn out something like the normal probability density function but its just turns into a mess when i try it. i am given the following information: \frac{\partial p}{\partial...

Hey, I need some guidance in this problem. Consider a rocket taking off vertically from rest in a gravitational field g, the equation of motion (which I had to derive in the previous part of this problem) is: m \dot{v} = -\dot{m}v_{ex} - mg where m is the mass of the rocket...

Slve by separation of variables \frac{\partial u}{\partial t} - k \frac{\partial^2 u}{\partial x^2} = 0 for 0 <x < pi, t > 0 u(0,t) = u(\pi,t) = 0 u(x,0)= \Sin^2 x let u (x,t) = X(x) T(t) \frac{X''}{X} = \frac{T'}{T} = -\lambda = \mu^2 also lambda must be positive...

we are given the laplacian: (d^2)u/(dx^2) + (d^2)u/(dy^2) = 0 where the derivatives are partial. we have the B.C's u=0 for (-1<y<1) on x=0 u=0 on the lines y=plus or minus 1 for x>0 u tends to zero as x tends to infinity. Using separation of variable I get the general solution u =...

I am to reduce the following PDE to 2 ODEs and find only the particular solutions: u_tt - u_xx - u = 0; u_t(x,0) = 0; u(0,t) = u(1,t) = 0 I guess u = X(x)T(t), and plug u_tt, u_xx into PDE and divide by u to get: T''/T = X''/X + 1 = K I solve X'' + (1-K)X = 0 first. From...

Hi, In physics, many PDEs' equation can be reduced to a number of ODEs' equations by "Separation of Variables". According to my knowledge, "Separation of Variables", it just subsitutes the function, for example F(x,y,z), by a product of three function, X(x)Y(y)Z(z). Why does it work? I mean...

two problems in particular, one i got in calc, the other in physics one asks if a = -v and v = 1 when t = 0 what is a possible position function for this equation the other one is given a = 3x and starting at rest from x = 0 find the velocity at 5 seconds i can't seem to...