2nd order with exponential and constant on right side

[ADGigus]

Hi everybody,

How do I solve this differential equation ??:

y'' = a(Exp(-b*y)-1) ;

where a, b are constants

with the boundaries conditions :

y'(x=0)=-K1
y'(x=L)=0

without the constant term I can do

y''*y' = y' a Exp(-b y)

then integrate it

$$\ {1/2} (y')^2= {a/b} ~Exp(-b y)$$

and so on and finaly find something in hyperbolic function Tanh()...

but With the constant term " -a " on the right side, I don't know how to start.

Thank very much for your help

 

[tiny-tim]

Welcome to PF!

Hi ADGigus! Welcome to PF!

easy-peasy …

∫ y' (a Exp(-b y) - a)

= ∫ay'Exp(-b y) - ∫ay'

 

[ADGigus]

ok but my problem is in the followings...

Thank you for your very quick answer and the welcome.

Yes I agree til there it's easy but it's the following that gives me troubles.

so
starting from :

$$y'' = a Exp(-b y) -a$$

$$y ' y'' = y ' a Exp(-b y) -a y '$$

after the integration on y on both sides, I have :

$$1/2~(y ')^2 = -a/b~ Exp(-b y) -a y + K$$$$y' =\sqrt{-2a (1/b ~ Exp(-b y)-y) + K}$$

and then I'm really stuck... :-(

 

[tiny-tim]

Hi ADGigus!

Yes … I'm stuck too.

Are you sure it's not y'' = a(Exp(-b*y-1)) ?

 

[ADGigus]

it's ok

Yes I'm sure it's correct,

it's coming from the expression of a Schottky diode (current density vs Voltage)

so it's :

J = a(Exp(-bV) -1)

because you want 0 current at bias of 0 Volts...

I already try many way to solve this equations, til now I'm fully stuck

Even numerically I don't find any correct way to solve it.

May be there is some polynomial approximation to use, but I don't really find anything fine.

if we take the developpement of exponential at the first order it's easy (1+V), but it's insufficient, I lose the diode physics and it does like it's something linear, so ohmic. And It's not enough for my calculation.