1. Consider the initial value problem (x + 5) y' + {2 / x-π) y= In x, y(1) = -10. Without solving the equation, what is the largest interval in which a unique solution is guaranteed to exist?
2. A fish tank is initially tilled with 400 liters of water containing 2g/liter of dissolved oxygen. At t 0. oxygenated water containing 10 g liter of oxygen flows in at n rate of 4 liter min. live well-mixed water is pumped out at a rate of 3 liter min from the tank. Write an initial value problem (do not solve) that describes. Q(t). the amount of dissolved oxygen in the tank at any time r > 0 (until the time when the tank, eventually, overflows).
3. Newton's Law of Cooling says that the rate at which the temperature of a body changes is proportional to the difference of the temperature of the body and the temperature of the surrounding medium. Suppose a cup of coffee at 100o C is placed in a room where the temperature is 25o C. Write an initial value problem (do not solve) to describe the temperature of the cup of coffee at any lime t > 0.