Final answer:
To determine the values of r for which the given differential equation has the solution of the form y = e^(rt), we substitute the solution into the equation and solve for r. The values of r that satisfy the equation are 2 and -15.
Step-by-step explanation:
To determine the values of r for which the given differential equation has the solution of the form y = e^(rt), we need to substitute this solution into the differential equation and find the values of r that satisfy the equation. The given differential equation is y" + 13y - 30y = 0. Substituting y = e^(rt) into the equation:
e^(rt)" + 13e^(rt) - 30e^(rt) = 0
Using the fact that the second derivative of y = e^(rt) is y" = r^2e^(rt), we can rewrite the equation as:
r^2e^(rt) + 13e^(rt) - 30e^(rt) = 0
Factoring out e^(rt), we have:
e^(rt)(r^2 + 13r - 30) = 0
For the equation to be satisfied, either e^(rt) = 0 or r^2 + 13r - 30 = 0. Since e^(rt) is never equal to zero, we only need to solve the quadratic equation:
r^2 + 13r - 30 = 0
To solve the quadratic equation, we can factor it as (r - 2)(r + 15) = 0. Therefore, the values of r that satisfy the given differential equation are r = 2 and r = -15.