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Does anybody know how to solve, analytically or numerically, the following differential equation :

[tex] \frac{d^2\Phi}{dx^2}-a.Sinh(\frac{\Phi}{U_{th}})=-b.Exp(-(\frac{x-x_{m}}{\sigma})^2})[/tex]

The unknown function is [tex]\Phi[/tex].

a and b are some strictly positive constants.

q[tex]\Phi[/tex] 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 [tex]BExp(-(\frac{x-x_{m}}{\sigma})^2})[/tex] the (non-uniform) dopant concentration in the substrate versus the distance to the silicon dioxide/silicon interface.

THANX.

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

Can you offer guidance or do you also need help?

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