Solve the differential equation $y ^ { \prime } = 5 y ( 1000 - y )$ subject to the initial condition $y ( 0 ) = 500$ . From your solution, and the value of the limit $\lim _ { t \rightarrow \infty } y ( t )$ .
A)5000
B)2500
C)1000
D)2000
E)200
F)20000
G)100
H)500

Question

Quizplus · Accepted Answer

This differential equation is a logistic growth equation. The solution to such equations typically approaches the carrying capacity as $t \rightarrow \infty$, which in this case is 1000. The carrying capacity is the value towards which the population (or in this case, $y$) stabilizes, given by the coefficient of $y$ in the term $(1000 - y)$.

Solve the Differential Equation y′=5y(1000−y)y ^ { \prime } = 5 y ( 1000 - y )y′=5y(1000−y)

Solve the Differential Equation $y ^ { \prime } = 5 y ( 1000 - y )$