The Method Of Separation Of Variables Can anybody explain me why the method of separation of variables  for linear homogeneous PDE works ? thanks 
 A: "Separation of variables" has two totally different meanings, which have nothing to do with each other.
(i) A general ODE is of the form $y'=f(x,y)$, where $f$ is an arbitrary function of two variables $x$, $y$. Sometimes this function is not so arbitrary, but is a product of two functions depending only on $x$, resp., on $y$:
$$y'=g(x) h(y)\ .$$ 
In this case writing formally $y'={dy\over dx}$ you can "separate the variables $x$ and $y$" and obtain the equation
$${1\over h(y)}\ dy\ =\ g(x)\ dx\ .$$
There is a certain formal procedure of integration on the left with respect to $y$ and on the right with respect to $x$ that finally produces all solutions of the given ODE. The proof that this works is not at all intuitive.
(ii) Given a linear homogeneous PDE in several variables, say: two variables $x$ and $t$ that together range over a maybe infinite rectangle in the $(x,t)$-plane, you can raise the question whether there are "special solutions" $f$ of the form $f(x,t)=X(x)\cdot T(t)$. For $f$ to be such a special solution it will be necessary that $X(\cdot)$ and $T(\cdot)$ satisfy certain linear ODEs which are easy to solve. When you are lucky you get a large bag full of such special solutions $f_\lambda(x,t)=X_\lambda(x)T_\lambda(t)$, $\ \lambda\in\Lambda$.
Now it is a fundamental feature of linear homogeneous systems of any kind that any linear combination of known solutions is again a solution. It follows that any function  $u$ of the form
$$u(x,t)=\sum_{\lambda\in\Lambda} c_\lambda X_\lambda(x)T_\lambda(t)$$
 is a solution of your PDE. Therefore it makes sense to try to determine the coefficients $c_\lambda$ in such a way that the other conditions (in particular, boundary conditions or initial conditions) present in your problem are met as well.
