Originally Posted by

**Sudharaka** Dear polo,

This differential equation could be solved in several ways. But I will give you the most easiest I can think of. In a differential equation if x is absent explicitily, (does not have $\displaystyle x^{1}$ terms.) this method can be used.

$\displaystyle \frac{d^2z}{dx^2}+z=bz^{3}$

Substitute, $\displaystyle p=\frac{dz}{dx}$

$\displaystyle \frac{dp}{dx}=\frac{dp}{dz}\times\frac{dz}{dx}=p\f rac{dp}{dz}$

$\displaystyle \frac{dp}{dx}+z=bz^{3}$

$\displaystyle p\frac{dp}{dz}+z=bz^{3}$

Seperation of variables and integrating yields, $\displaystyle \frac{p^2}{2}=\int{(bz^{3}-z)dz}$

$\displaystyle \frac{p^2}{2}=\frac{bz^{4}}{4}-\frac{z^{2}}{2}+C~;Where~C~is~an~arbitary~constant .$

Hope this will help you.