Final answer:
The values of r for which the given differential equation has solutions of the form y = eʳᵗ are r = -3 and r = 2.
Step-by-step explanation:
To determine the values of r for which the given differential equation has solutions of the form y = eʳᵗ, we can substitute the solution into the differential equation and solve for r.
Substituting y = eʳᵗ into the differential equation, we get (r² + r - 6)eʳᵗ = 0. Since (r² + r - 6)eʳᵗ = 0 represents a product of two factors, either (r² + r - 6) = 0 or eʳᵗ = 0.
Simplifying (r² + r - 6) = 0, we can factor it as (r + 3)(r - 2) = 0. Therefore, the values of r that the given differential equation has solutions of the form y = eʳᵗ are r = -3 and r = 2.