Hello,
I'm trying to solve symbolically the following second-order differential equation using the DSolve function.
My expression contains a number of variables, but it is of the form:
$f''(x) = a \exp{\frac{b-f(x)}{c}} + d $
Because
$ f'' $ contains an exponential, I expect the solution
$f(x)$ to also contain an exponential. However, the DSolve function provides me with a quadratic solution, which is not what I am looking for.
Am I misusing the DSolve function? Have I misunderstood the mathematics? Any help is appreciated.
In[13]:= q = 1.602 * 10 ^ -19 (* C *)
e0 = 8.854 * 10 ^ -14 (* F/cm *)
en = 13.9
Ncn = 2.1 * 10 ^ 17 (* cm^-3 *)
Fn = -0.079 (* eV, = -12.656 * 10 ^ -21 J*)
xN = 1 * 10 ^ -4 (* cm *)
Nd = 10 ^ 16 (* cm^-3 *)
ND = 2 * 10 ^ 18 (* cm^-3 *)
kT = 0.0259 (* eV *)
(*f[x] = Ec[x], conduction band energy*)
(* DSolve with no given boundary conditions *)
DSolve[f''[x]== (-q^2)/(e0*en) * (Ncn*Exp[(Fn-f[x])/kT] - Nd),f[x],x]
Out[13]= 1.602*10^-19
Out[14]= 8.854*10^-14
Out[15]= 13.9
Out[16]= 2.1*10^17
Out[17]= -0.079
Out[18]= 1/10000
Out[19]= 10000000000000000
Out[20]= 2000000000000000000
Out[21]= 0.0259
Out[22]= {{f[x]->1.04266*10^-10 (-2.29963*10^19 C[1]+1. (x+C[2])^2)}}
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