Find all constants r for which y erx is a solution of y 5y

Find all constants r for which y = e^rx is a solution of y\" - 5y\' - 6y = 0.

Solution

Given y = e^rx

y\' = r e^rx

y\" = r² e^rx

we know that y\" - 5y\' -6y =0

r² e^rx - 5 r e^rx - 6 e^rx =0

e^rx (r^2 -5r - 6) = 0

r^2 -5r - 6 = 0

r^2 - 2r - 3r -6 = 0

r( r- 2) -3 ( r- 2 ) = 0

therefore, the constants of r = 2 , 3