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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 :

Answer:

A differential equation of the form: [tex]dy/dx = f(x, y)[/tex] is [tex]\frac{-x}{y+2}[/tex]

Step-by-step explanation:

Step 1 of 3

a) Acceleration[tex]$\rightarrow v^{\prime}(t)$[/tex]

velocity [tex]$\rightarrow v(t)$[/tex]

Given that [tex]$v^{\prime}(t) \propto v^{2}(t)$[/tex]

[tex]\frac{d v}{d t}=k v^{2}[/tex]

[tex]k[/tex]  is proportionality constant.

Step 2 of 3

b) The population of Zootopia in year [tex]$t$[/tex], is [tex]$P(t)$[/tex]

Let [tex]$\beta$[/tex] the 'birth rate of Animals and [tex]$\delta$[/tex] the death rate of the population.

Then the population model is

[tex]\frac{d p}{d t}=(\beta-\delta) P$[/tex]

Given that [tex]$\beta=50000$[/tex] and

[tex]$\delta=40000$[/tex]

So, [tex]$\frac{d P}{d t}=10000 P$[/tex]

Step 3 of 3

c) First we take the derivative of [tex]$f(x)$[/tex] i. e [tex]$f^{\prime}(x)$[/tex].

The evaluate it at [tex]$m_{1}=f^{\prime}(2)[/tex]

[tex]m_{2}=\frac{-1}{f^{\prime}(2)}$[/tex]

The normal line with slope [tex]$\frac{-1}{f^{\prime}(2)}$[/tex]

[tex]y-(-2)=\frac{-1}{f^{\prime}(2)}(x-0) \\[/tex]

[tex]y+2=\frac{-x}{f^{\prime}(2)}[/tex]

[tex]f^{\prime}(2)=\frac{f x}{(y+2)} \\[/tex]

[tex]&f^{\prime}(2)+\frac{x}{(y+2)}=0[/tex]

[tex]\frac{d y}{d x}=\frac{-x}{y+2}[/tex]

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