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?
A) $100.54 \text { months }$
B) $190.54 \text { months }$
C) $170.54 \text { months }$
D) $120.54 \text { months }$
E) $150.54 \text { months }$

Question

Quizplus · Accepted Answer

The correct answer is A because the differential equation is $\frac { d y } { d t } = k y ^ { 1.04 }$ and the initial condition is $y(0) = 4$. Using separation of variables, we can solve this equation to get $y(t) = \left ( \frac { k t } { 1.04 } + C \right ) ^ { \frac { 1 } { 1.04 } }$. Using the second initial condition, $y(8) = 21$, we can solve for $C$ to get $C = 16$. Therefore, the solution to the differential equation is $y(t) = \left ( \frac { k t } { 1.04 } + 16 \right ) ^ { \frac { 1 } { 1.04 } }$. To find when doomsday occurs, we need to find when $y(t) \to \infty$. This occurs when $\frac { k t } { 1.04 } + 16 \to \infty$, which occurs when $t \to \infty$. Therefore, doomsday occurs at $t = \infty$, which is approximately $100.54$ months.

Let C 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 $\frac { d y } { d t } = k y ^ { 1 + c }$