Correct Answer  Option 4 : n(x) = K exp(x/L)
Concept:
The general solution of different types of roots for a particular DE is shown below:
Roots of AE

Nature

General solution

a, b, c ⋯

Real & distinct

y(x) = C_{1}e^{ax} + C_{2}e^{bx} + ⋯

a, a, a, a ⋯

Real & repeated

y(x) = C_{1}e^{ax} + C_{2}xe^{ax} + C_{3} x^{2}e^{ax} +⋯

α ± iβ

Complex

y(x) = e^{αx}[C_{1 }cos βx + C_{2 }sin βx]

α ± iβ
n times

Complex & repeated

y(x) = e^{αx}[(C_{1}+C_{2}x + ⋯ C_{n}x^{n1}) cos βx + D_{1}+D_{2}x + ⋯ D_{n}x^{n1}) sin βx ]

± iβ

purely imaginary

y(x) = C_{1 }cos βx + C_{2 }sin βx

irrational
α ± √β

Irrational

y(x) = e^{αx}[C_{1 }cosh √βx + C_{2 }sinh √βx]

Calculation:
Given DE is:
\(\frac{{{d^2}n\left( x \right)}}{{d{x^2}}}  \frac{{n\left( x \right)}}{{{L^2}}} = 0 \) (1)
n(0) = k (2)
n(∞) = 0 (3)
From equation (1) we can write the DE as:
\(\left( {{D^2}  \frac{1}{{{L^2}}}} \right)n = 0\)
The auxiliary equation is:
\({D^2}  \frac{1}{{{L^2}}} = 0\)
D = ± 1/L
Roots are real & distinct
The general solution is:
n(x) = c_{1}e^{x/L} + c_{2}e^{x/L} (4)
\(\frac{{dn\left( x \right)}}{{dx}} = \frac{1}{L}{c_1}{e^{\frac{x}{L}}}  \frac{{{c_2}}}{L}{e^{  \frac{x}{L}}}\) (5)
Using equations (2), (4) becomes
K = c_{1} + c_{2} (6)
Using (3), (4) becomes
0 = c_{1}(∞) + c_{2}(0)
c_{1}(∞) = 0
c_{1} = 0
From (6)
c_{2} = k
The solution of the equation (1) is:
n(x) = ke^{x/L}