For each of the following scenarios, provide a reasonable IVP model. Let $$t$$ represent time, let $$z$$ represent the value given in the scenario, and use uppercase letters for positive constants. Then label each term and initial value of each IVP to describe what it represents from the scenario.

A glass of sweet tea is left outside on a summer day. After some time, ice is added to the glass, chilling the drink at a constant rate. Let z measure the temperature of the tea over time, assuming the outside temperature is kept constant.
An object is released from rest from above the ground. Let z measure its upward velocity at a given time. Assume quadratic air resistance.
A mass is attached to a spring. The mass is compressed inward from the spring's natural position, then released from rest. Assume no damping or friction. Let z measure the outward displacement of the mass from its natural position on the spring.
Suppose the population of a species follows the logistical model: births are linear based on population, and deaths are quadratic based on population. Let z measure the population of this species at a given time.

Then, show that your model is equivalent to one of the following IVP models by relabeling constants and using algebra as needed. (It's possible that more than one IVP may be used for a scenario. If so, choose any of them.)

$D {z''} = -M z \hspace{2em}z(0)=- W ,z'(0)=0$
$0 = Q z^{2} - S z + {z'} \hspace{2em}z(0)= D$
$0 = V z + E {z'} - K \hspace{2em}z(0)= J$
$-E V - E {z'} = -M z^{2} \hspace{2em}z(0)=0$
$\frac{B z}{D} + {z'} = {\left(A - Y\right)} B \mathrm{u}\left(-L + t\right) + B Y \hspace{2em}z(0)= W$
${z'} = L W - W z - M \mathrm{u}\left(-Y + t\right) \hspace{2em}z(0)= S$

Answer:

An object is released from rest from above the ground. Let z measure its upward velocity at a given time. Assume quadratic air resistance.
- $-E V - E {z'} = -M z^{2} \hspace{2em}z(0)=0$
A solution of salt water is pumped into a tank of less salty water, while mixed water is pumped out at the same rate. After a certain amount of time, the concentration of salt flowing into the tank is instantly increased. Let z measure the mass of salt in the tank at a given time.
- $\frac{B z}{D} + {z'} = {\left(A - Y\right)} B \mathrm{u}\left(-L + t\right) + B Y \hspace{2em}z(0)= W$
Suppose the population of a species follows the logistical model: births are linear based on population, and deaths are quadratic based on population. Let z measure the population of this species at a given time.
- $0 = Q z^{2} - S z + {z'} \hspace{2em}z(0)= D$
A mass is attached to a spring. The mass is compressed inward from the spring's natural position, then released from rest. Assume no damping or friction. Let z measure the outward displacement of the mass from its natural position on the spring.
- $D {z''} = -M z \hspace{2em}z(0)=- W ,z'(0)=0$
A glass of sweet tea is left outside on a summer day. After some time, ice is added to the glass, chilling the drink at a constant rate. Let z measure the temperature of the tea over time, assuming the outside temperature is kept constant.
- ${z'} = L W - W z - M \mathrm{u}\left(-Y + t\right) \hspace{2em}z(0)= S$
A circuit includes a battery providing constant voltage, a resistor, and an inductor. Assume some initial current is flowing, and let z measure the current throughout this circuit at a given time.
- $0 = V z + E {z'} - K \hspace{2em}z(0)= J$