# Numerical resolution of 2nd order non-linear differential equation

1. Oct 28, 2004

### zeus_the_almighty

Hi Everybody,
Does anybody know how to solve, analytically or numerically, the following differential equation :
$$\frac{d^2\Phi}{dx^2}-a.Sinh(\frac{\Phi}{U_{th}})=-b.Exp(-(\frac{x-x_{m}}{\sigma})^2})$$

The unknown function is $$\Phi$$.
a and b are some strictly positive constants.
q$$\Phi$$ is the energy band bending of a P-type substrate MOS capacitor versus the distance to the silicon dioxide/silicon interface.

Uth is the thermal voltage and $$BExp(-(\frac{x-x_{m}}{\sigma})^2})$$ the (non-uniform) dopant concentration in the substrate versus the distance to the silicon dioxide/silicon interface.

THANX.