Final answer:
To find the values of r such that y = e^t satisfies the differential equation y" - 11y' + 30y = 0, substitute y = e^t into the equation and solve for r.
Step-by-step explanation:
To find the values of r such that y = e^t satisfies the differential equation y" - 11y' + 30y = 0, we need to substitute the given expression for y into the differential equation and solve for r.
Substituting y = e^t into the equation, we get (e^t)" - 11(e^t)' + 30(e^t) = 0.
Simplifying further, e^t - 11e^t + 30e^t = 0. Combining like terms, we have -10e^t + 30e^t = 0. Factoring out e^t, we get e^t(-10 + 30) = 0. Solving for r, we find r = 1.