Find the orthogonal trajectories of the given family (represented by the purple curves in the plot below): $$y = C e^{-x}.$$ Write your answer as $f(y) = g(x) + C$, where both $f(y)$ and $g(x)$ vanish at the origin. Moreover, you should multiply by constants if necessary so that $f(1) = 1/2$ and $g(1) = 1$. $$\answer {\frac {1}{2} y^2} = \answer {x} + C.$$

Find the orthogonal trajectories of the given family (represented by the purple curves in the plot below): $$y = x - \frac {x^3}{3} + C.$$ Write your answer as $y = f(x) + C$, where $f(0) = 0$ $$y = \answer {\frac {1}{2} \ln \left | \frac {1-x}{1+x} \right |} + C.$$

Find the orthogonal trajectories of the given family: $$y = \tan \left ( C + x + \frac {x^3}{3} \right )$$ Write your answer as $f(y) = g(x) + C$, where both $f(y)$ and $g(x)$ vanish at the origin. Moreover, you should multiply by constants if necessary so that $f(1) = 4/3$ and $g(1) = -\pi /4$. $$\answer {y + \frac {y^3}{3}} = \answer {- \arctan x} + C$$

It takes about three days to defrost a 10 pound imitation frozen turkey in a home refrigerator. Suppose that the initial temperature is $-16^\circ$C and the ambient temperature is $2^\circ$C inside the refrigerator. If after exactly two days, the temperature of the frozen turkey is $0^\circ$C, find a formula for the temperature $y(t)$ (in degrees Celsius) as a function of time for all $t > 0$, where $t$ is measured in days $$y = \answer { -18 e^{-t \ln 3} + 2}.$$

At time $t = 0$, a cup of tea has temperature $46^\circ$C. Ten minutes later, its temperature is $34^\circ$C. Ten minutes after that, the temperature is then $28^\circ$C. Assuming the temperature obeys Newton’s Law of cooling, what is the temperature (in degrees Celsius) as a function of time $y(t)$ for all $t > 0$, where $t$ is measured in minutes? $$y(t) = \answer {24 e^{-t \ln (2)/10} + 22}$$

A 1000 liter tank is filled with a sugar solution: 200 kilograms of sugar dissolved in 1000 liters of pure water. The solution in the tank is kept thoroughly mixed at all times. At time $t = 0$, the attendants begin adding 1 liter per minute of dissolved sugar at a concentration of 0.1 kilograms per liter. At the same time, One liter per minute is drained from the tank to keep the overall volume of solution in the tank at 1000 liters. Find a function $S(t)$ for the total amount of sugar in the tank (in kilograms) a time $t > 0$, where $t$ is measured in minutes. $$S(t) = \answer { 100 + 100 e^{-t/1000}}.$$

A small flask contains 10mL of solvent in which 1 gram of substance X is initially dissolved. The solution is slowly drained at a rate of 3mL per hour while 4mL per hour of pure solvent is added (with the whole solution being kept thoroughly mixed). Find a formula $X(t)$ for the amount of substance $X$ (in grams) sill in the beaker at time $t$, where $t$ is measured in hours. $$X(t) = \answer { \frac {1000}{(t+10)^3}}.$$