Consider the differential equation y'' − y' − 20y = 0. Verify that the functions e−4x and e5x form a fundamental set of solutions of the differential equation on the interval (−[infinity], [infinity]). The functions satisfy the differential equation and are linearly independent since the Wronskian W e−4x, e5x =

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

                  ≠0

Therefore are linearly independent.[ ∵W≠0]