Describe how solutions appear to behave as t increases and how their behavior depends on the initial value yo when ! y' = y(8 – ty) 8 For yo > 0 solutions decrease until they intersect the curve y =, then they increase. For yo = 0,y = 0 is an equilibrium solution. For yo < 0, solutions decrease without bound as t → 0. 8 For yo > 0 solutions increase until they intersect the curve y =, then they decrease. For yo = 0,y = 0 is an equilibrium solution. For yo < 0, solutions increase without bound as t → 0. 8 For yo > 0 solutions decrease until they intersect the curve y =, then they increase. For yo = 0,y = 0 is an equilibrium solution. For yo <0, solutions increase without bound as t → 0.
Describe how solutions appear to behave as t increases and how their behavior depends on the initial value yo when ! y' = y(8 – ty) 8 For yo > 0 solutions decrease until they intersect the curve y =, then they increase. For yo = 0,y = 0 is an equilibrium solution. For yo < 0, solutions decrease without bound as t → 0. 8 For yo > 0 solutions increase until they intersect the curve y =, then they decrease. For yo = 0,y = 0 is an equilibrium solution. For yo < 0, solutions increase without bound as t → 0. 8 For yo > 0 solutions decrease until they intersect the curve y =, then they increase. For yo = 0,y = 0 is an equilibrium solution. For yo <0, solutions increase without bound as t → 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**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_](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77cac1a6-5ad9-4f72-bdd9-21a202e53df4%2F808fb194-91ff-4a12-9811-abb08843f66f%2Fmultxc9_processed.png&w=3840&q=75)
Transcribed Image Text:**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_
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