Question

Write out the differential equations, DO NOT solve, for the following word problems.

a. The acceleration, v'(t), of a coasting motorboat is proportional to the square of its velocity, v(t).
b. The population of Zootopia in year t, is P(t): Animals are born at a rate propor­tional to the population. The death rate is also proportional to the population. Animals move to Zootopia at a rate of 50,000 animals a year, and 40,000 animals move away in a year.
c. A function g(x) is described by the following property of its graph: At each point (x, y) on the graph y=g(x), the normal line to the graph passes through the point (0, -2). Use this information to find a differential equation of the form: dy/dx = f(x, y) Having the function g(x) as its solution.

Answer 1

A differential equation of the form:
`dy/dx = f(x, y)` is
`(-x)/(y+2)`

Explanation:

Step 1 of 3

a) Acceleration
`$\rightarrow v^(\prime)(t)$`

velocity
`$\rightarrow v(t)$`

Given that
`$v^(\prime)(t) \propto v^(2)(t)$`

`(d v)/(d t)=k v^(2)`

`k` is proportionality constant.

Step 2 of 3

b) The population of Zootopia in year
`$t$`, is
`$P(t)$`

Let
`$\beta$` the 'birth rate of Animals and
`$\delta$` the death rate of the population.

Then the population model is

`(d p)/(d t)=(\beta-\delta) P$$`

Given that
`$\beta=50000$` and

`$\delta=40000$`

So,
`$(d P)/(d t)=10000 P$`

Step 3 of 3

c) First we take the derivative of
`$f(x)$` i. e
`$f^(\prime)(x)$`.

The evaluate it at
`$m_(1)=f^(\prime)(2)`

`m_(2)=(-1)/(f^(\prime)(2))$`

The normal line with slope
`$(-1)/(f^(\prime)(2))$`

`y-(-2)=(-1)/(f^(\prime)(2))(x-0) \\`

`y+2=(-x)/(f^(\prime)(2))`

`f^(\prime)(2)=(f x)/((y+2)) \\`

`&f^(\prime)(2)+(x)/((y+2))=0`

`(d y)/(d x)=(-x)/(y+2)`