Exam 9: Differential Equations

A phase trajectory is shown for populations of rabbits $( R )$ and foxes $( F )$ . Describe how each population changes as time goes by. Select the correct statement.

Let c be a positive number. A differential equation of the form $\frac { d y } { d t } = k y ^ { 1 + c }$ where k is a positive constant is called a doomsday equation because the exponent in the expression $k y ^ { 1 + c }$ is larger than the exponent 1 for natural growth. An especially prolific breed of rabbits has the growth term $k y ^ { 1.04 }$ . If $4$ such rabbits breed initially and the warren has $21$ rabbits after $8$ months, then when is doomsday?

The Pacific halibut fishery has been modeled by the differential equation $\frac { d y } { d t } = k y \left( 1 - \frac { y } { K } \right)$ where $y ( t )$ is the biomass (the total mass of the members of the population) in kilograms at time t (measured in years), the carrying capacity is estimated to be $M = 40 \times 10 ^ { 4 } \mathrm {~kg}$ and $k = 0.02$ per year. If $y ( 0 ) = 5 \times 10 ^ { 4 }$ , find the biomass a year later.

Find the solution of the differential equation that satisfies the initial condition $y ( 0 ) = 6$ . $\frac { d y } { d x } = 6 x ^ { 5 } y$

Find the orthogonal trajectories of the family of curves. $y = ( x + k ) ^ { - 8 }$

Which of the following functions are the constant solutions of the equation $\frac { d y } { d t } = y ^ { 4 } - 6 y ^ { 3 } + 8 y ^ { 2 }$ a. $y ( t ) = - 4$ b. $y ( t ) = e ^ { t }$ c. $y ( t ) = - 2$ d. $y ( t ) = 0$ e. $y ( t ) = 5$

A certain small country has $20 billion in paper currency in circulation, and each day $70 million comes into the country's banks. The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks. Let $x = x ( t )$ denote the amount of new currency in circulation at time t with $x ( 0 ) = 0$ . Formulate and solve a mathematical model in the form of an initial-value problem that represents the "flow" of the new currency into circulation (in billions per day).