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 5 such rabbits breed initially and the warren has 23 rabbits after 5 months, then when is doomsday? Select the correct answer.
A) $138.06$ months
B) $158.06$ months
C) $118.06$ months
D) $88.06$ months
E) $68.06$ months

Question

Quizplus · Accepted Answer

The solution to the differential equation is $ y(t) = \left( \frac{1}{1 - c} \right) \left( \frac{1}{y_0^{c} - ktc} \right)^{1/c} $. Solving for $ t $ when $ y(t) \to \infty $ gives the doomsday time. Using the given values, the doomsday occurs at approximately $ 68.06 $ 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 $\frac { d y } { d t } = k y ^ { 1 + c }$