Final answer:
To find the values of r for which the given differential equation has solutions of the form y = eʳᵗ, we need to substitute y into the differential equation and solve for r.
Step-by-step explanation:
To determine the values of r for which the given differential equation has solutions of the form y = eʳᵗ, we need to substitute y into the differential equation and solve for r. Let's start by finding the first and second derivatives of y. The first derivative is y' = rᵗ eʳქ and the second derivative is y'' = rˣ eʳქ. Substituting these values into the differential equation, we get rˣ eʳქ + rᵗ eʳქ - 6eʳᵗ = 0. Factoring out eʳᵗ, we get eʳᵗ(rˣ + rᵗ - 6) = 0. To find the values of r, we set the expression inside the parentheses equal to zero and solve the resulting equation: rˣ + rᵗ - 6 = 0.