Answer:
Therefore the auxiliary solution is 
Therefore
are linearly independent
Step-by-step explanation:
Given, the differential equation is
y"-y'-20 y=0
Let
be the solution of the above differential equation.
y'=
and 
Then the above differential equation becomes







If two roots of m are real and distinct then the auxiliary solution is
[where a and b are two roots of m]
Therefore the auxiliary solution is 
Wronskian
![W(e^{-4x},e^{5x})=\left[\begin{array}{cc}e^{-4x}&e^{5x}\\-4e^{-4x}&5e^{5x}\end{array}\right]](https://tex.z-dn.net/?f=W%28e%5E%7B-4x%7D%2Ce%5E%7B5x%7D%29%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7De%5E%7B-4x%7D%26e%5E%7B5x%7D%5C%5C-4e%5E%7B-4x%7D%265e%5E%7B5x%7D%5Cend%7Barray%7D%5Cright%5D)

≠0
Therefore
are linearly independent.[ ∵W≠0]