Determine the values of r for which the given differential e

Determine the values of r for which the given differential equation has solutions of the form y = t^r for t > 0. (Enter your answers as a comma-separated list.) t^2y\" + 10fy\' + 20y = 0

Solution

Let y = t^r

=> y\' = rt^r-1

=> y\'\' = r(r - 1)t^r-2

Plugging above in the differential equation

=> t².r(r - 1)t^r-2 + 10t.rt^r-1 + 20.t^r = 0

=> t^r( r(r - 1) + 10r + 20 ) = 0

=> r² + 9r + 20 = 0

=> ( r + 5 )( r + 4 ) = 0

=> r = -5 , -4