( Find the general solution to the homogeneous differential equation The solution has the form d²y dt² - dy 12- + 100y = 0 dt y = C₁f₁(t) + C₂f₂(t) with f₁(t) = f₂ (t) = Left to your own devices, you will probably write down the correct answers, but in case you want to quibble, enter your answers so that the functions are normalized with their values at t = 0 equal to 0 and 1(respectively), and they are expressed as simply as possible. and

Newton's Law of Cooling states that the rate at which the temperature of a cooling object decreases and the rate at which a warming object increases are proportional to the difference between the temperature of the object and the temperature of the surrounding medium. The differential equation formula is given as: dT = k (T – A) dt - T = temperature of object at time t А ambient temperature (aka temperature of the surrounding medium) k = constant of proportionality Okay, now here's the problem. A glass of delicious orange soda (with an initial temperature of 34°F) is placed in a room with a constant temperature of 72°F. One hour later, the temperature of the orange soda is 52°F. Find the exact simplified value of k. Hint: solve the differential equation for T and note that A is a constant.

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