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.

An object is thrown upward from above the ground. Let z measure its altitude at a given time. Assume linear air resistance.
Suppose the population of a species generally follows the logistical model: births are linear based on population, and deaths are quadratic based on population. However, at some point in the model, a natural disaster instantly wipes out a fraction of the population. Let z measure the population of this species at a given time.
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.
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.

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.)

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

Answer:

An object is thrown upward from above the ground. Let z measure its altitude at a given time. Assume linear air resistance.
- $-M W - M {z''} = A {z'} \hspace{2em}z(0)= P ,z'(0)= J$
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.
- $D K = {\left(D - J\right)} K \mathrm{u}\left(-W + t\right) + \frac{K z}{P} + {z'} \hspace{2em}z(0)= E$
Suppose the population of a species generally follows the logistical model: births are linear based on population, and deaths are quadratic based on population. However, at some point in the model, a natural disaster instantly wipes out a fraction of the population. Let z measure the population of this species at a given time.
- $-A z^{2} - M \delta\left(-K + t\right) - {z'} = -R z \hspace{2em}z(0)= N$
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.
- $0 = -Q z - M {z''} \hspace{2em}z(0)=- W ,z'(0)=0$
A bowl of hot soup sits in a colder room. Assume the room's temperature is kept constant, and let z measure the temperature of the soup over time.
- $-K z - {z'} = -K N \hspace{2em}z(0)= P$
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.
- $K z + E {z'} = R \hspace{2em}z(0)= B$