
Separation of variables
hi all,
im practising separation of variables for partial differential equations. i came across this site which says:
2. Separation of Variables
'NOTE: In this variables separable section we only deal with first order, first degree differential equations.'
so , if i have a second order or above PDE, then i cant use separation of variables?
i thought one can use separation of variables for example in the wave equation!! that is a second order PDE....
ok im confused (Doh)
can anyone help me out?
i hope i am clear with this question problem
thank you all very much(Itwasntme)

I think you and the site are talking about different things.
You are talking about separating independent variables in a partial differential equation:
i.e. solving $\displaystyle \frac{\partial^2 \phi}{\partial x^2}= \frac{\partial^2\phi}{\partial t^2}$ by writing $\displaystyle \phi(x, t)= X(x)T(t)$, the product of two functions, each a function of one of the independent variables.
The site you cite is talking about separating the independent and dependent variables in an ordinary differential equation:
$\displaystyle \frac{dy}{dx}= xy x^2y= y(x x^2)$
which can be written as $\displaystyle \frac{dy}{y}= (x x^2)dx$.
That latter can only be done with first order equations.

For certain types of second order DE You can use separation of variables. An example is...
$\displaystyle y^{''} = f(y)$ (1)
With the substitution...
$\displaystyle \displaystyle y^{''} = \frac{d y^{'}}{dy} \ \frac{d y}{dx} = y^{'}\ \frac{d y^{'}}{dy}$ (2)
... the (1) becomes the first order separate variables DE...
$\displaystyle \displaystyle y^{'}\ \frac{d y^{'}}{dy}= f(y)$ (3)
... the solution of which is...
$\displaystyle \displaystyle y^{'} = \sqrt {2\ \phi (y) + c_{1}}$ (4)
... where $\displaystyle \phi(y) $ is a primitive of $\displaystyle f(y)$. Now if in (4) You separe the variables again You obtain...
$\displaystyle \displaystyle dx = \frac{dy}{\sqrt {2\ \phi (y) + c_{1}}}$ (5)
... so that the general solution of (1) is...
$\displaystyle \displaystyle x= \int \frac{dy}{\sqrt {2\ \phi (y) + c_{1}}} + c_{2}$ (6)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$