# Thread: Separation of variables

1. ## 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
can anyone help me out?
i hope i am clear with this question problem

thank you all very much

2. 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 $\frac{\partial^2 \phi}{\partial x^2}= \frac{\partial^2\phi}{\partial t^2}$ by writing $\phi(x, t)= X(x)T(t)$, 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:
$\frac{dy}{dx}= xy- x^2y= y(x- x^2)$
which can be written as $\frac{dy}{y}= (x- x^2)dx$.

That latter can only be done with first order equations.

3. For certain types of second order DE You can use separation of variables. An example is...

$y^{''} = f(y)$ (1)

With the substitution...

$\displaystyle y^{''} = \frac{d y^{'}}{dy} \ \frac{d y}{dx} = y^{'}\ \frac{d y^{'}}{dy}$ (2)

... the (1) becomes the first order separate variables DE...

$\displaystyle y^{'}\ \frac{d y^{'}}{dy}= f(y)$ (3)

... the solution of which is...

$\displaystyle y^{'} = \sqrt {2\ \phi (y) + c_{1}}$ (4)

... where $\phi(y)$ is a primitive of $f(y)$. Now if in (4) You separe the variables again You obtain...

$\displaystyle dx = \frac{dy}{\sqrt {2\ \phi (y) + c_{1}}}$ (5)

... so that the general solution of (1) is...

$\displaystyle x= \int \frac{dy}{\sqrt {2\ \phi (y) + c_{1}}} + c_{2}$ (6)

Kind regards

$\chi$ $\sigma$