Please solve & show steps... **Explanation of Solution Behavior for the Differential Equation**Consider the differential equation:\[ y' = y(8 - ty) \]We are interested in describing how solutions appear to behave as \( t \) increases and how their behavior depends on the initial value \( y_0 \) when \( t = 0 \).### Options:1. **Option A:** For \( y_0 > 0 \), solutions decrease until they intersect the curve \( y = \frac{8}{t} \), then they increase. For \( y_0 = 0 \), \( y = 0 \) is an equilibrium solution. For \( y_0 < 0 \), solutions decrease without bound as \( t \to \infty \).2. **Option B:** For \( y_0 > 0 \), solutions increase until they intersect the curve \( y = \frac{8}{t} \), then they decrease. For \( y_0 = 0 \), \( y = 0 \) is an equilibrium solution. For \( y_0 < 0 \), solutions increase without bound as \( t \to \infty \).3. **Option C:** For \( y_0 > 0 \), solutions decrease until they intersect the curve \( y = \frac{8}{t} \), then they increase. For \( y_0 = 0 \), \( y = 0 \) is an equilibrium solution. For \( y_0 < 0 \), solutions increase without bound as \( t \to \infty \).4. **Option D:** For \( y_0 > 0 \), solutions increase until they intersect the curve \( y = \frac{8}{t} \), then they decrease. For \( y_0 = 0 \), \( y = 0 \) is an equilibrium solution. For \( y_0 < 0 \), solutions decrease without bound as \( t \to \infty \).5. **Option E:** For \( y_0 > 0 \), solutions increase without bound as \( t \to \infty \). For \( y_0 = 0 \), \( y = 0 \) is an equilibrium solution. For \( y_

Name: Please solve & show steps... **Explanation of Solution Behavior for th
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: Please solve & show steps... **Explanation of Solution Behavior for the Differential Equation**Consider the differential equation:\[ y' = y(8 - ty) \]We are interested in describing how solutions appear to behave as \( t \) increases and how their be