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 $k y ^ { 106 }$. If 3 such rabbits breed initially and the warren has 27 rabbits after 3 months, then when is doomsday?
A) $47.03$ months
B) $97.03$ months
C) $67.03$ months
D) $137.03$ months
E) $117.03$ months

Question

Quizplus · Accepted Answer

The solution to the differential equation $\frac{dy}{dt} = ky^{106}$ is $y(t) = \left(\frac{1}{1 - 105kt}\right)^{1/105}$. Given $y(0) = 3$ and $y(3) = 27$, we can find $k$ and then solve for the time $t$ when $y(t)$ becomes infinite, which is the doomsday. Solving gives approximately $47.03$ months.

Let ccc Be a Positive Number dydt=ky1+c\frac { d y } { d t } = k y ^ { 1 + c }dtdy​=ky1+c

Let $c$ Be a Positive Number $\frac { d y } { d t } = k y ^ { 1 + c }$