Problem 7. A researcher was studying a small forest with population of foxes and rabbits. After several years of observations they concluded that the population of rabbits oscillate around 1000 rabbits and the population of foxes oscillates around 100 foxes. To model that they wrote the population of rabbits as P₁ = 1000+y₁ and the population of foxes as P₂ : 100+ y2. = (a) Pick which of the following four differential equations for y₁, y2 best models the above situation. A), y'= (22) y, B) y' = ({ 02 c'), y' = ( 0-2 (112²) y 0 -4 1 1 ¹) y, D)y' = 4) ✓ = (²₂) y -4 0 1 B) as this is the only one with purely imaginary eigenvalues, corresponding to stable circle point. (b) For equation B) and initial condition y₁ (0) = 0, y2 (0) = 10, what is the maximal amount of rabbits predicted by this model over time? The solution of the equation gives y₁ (t) = -√√2 × 10 sin(√2t) and y₂(t) = 10 cos (√2t). so the maiximal value of y₁ is √2× 10. The maximal value of rabbit population, as predicted by this model, is then 1000 + √2 × 10.
Problem 7. A researcher was studying a small forest with population of foxes and rabbits. After several years of observations they concluded that the population of rabbits oscillate around 1000 rabbits and the population of foxes oscillates around 100 foxes. To model that they wrote the population of rabbits as P₁ = 1000+y₁ and the population of foxes as P₂ : 100+ y2. = (a) Pick which of the following four differential equations for y₁, y2 best models the above situation. A), y'= (22) y, B) y' = ({ 02 c'), y' = ( 0-2 (112²) y 0 -4 1 1 ¹) y, D)y' = 4) ✓ = (²₂) y -4 0 1 B) as this is the only one with purely imaginary eigenvalues, corresponding to stable circle point. (b) For equation B) and initial condition y₁ (0) = 0, y2 (0) = 10, what is the maximal amount of rabbits predicted by this model over time? The solution of the equation gives y₁ (t) = -√√2 × 10 sin(√2t) and y₂(t) = 10 cos (√2t). so the maiximal value of y₁ is √2× 10. The maximal value of rabbit population, as predicted by this model, is then 1000 + √2 × 10.
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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Question
can you explain how they got these answers for a & b
Transcribed Image Text:Problem 7. A researcher was studying a small forest with population of foxes and rabbits. After
several years of observations they concluded that the population of rabbits oscillate around 1000
rabbits and the population of foxes oscillates around 100 foxes. To model that they wrote the
population of rabbits as P₁ 1000+y₁ and the population of foxes as P₂ = 100+ y₂.
=
(a) Pick which of the following four differential equations for y₁, y2 best models the above situation.
22
0
-2
MY-(32)x. BY-(17),
A), y' = )y, B)y':
=
02
-4
20
CLY (1¹)Y. DIY-(8)
C), y' =
y, D)y' =
-4
0
1
y
B) as this is the only one with purely imaginary eigenvalues, corresponding to stable circle
point.
(b) For equation B) and initial condition y₁(0) = 0, y2(0) = 10, what is the maximal amount of
rabbits predicted by this model over time?
The solution of the equation gives y₁ (t) = -√√2 × 10 sin(√2t) and y₂(t) = 10 cos(√2t). so the
maiximal value of y₁ is √2 × 10. The maximal value of rabbit population, as predicted by this
model, is then 1000 + √2 × 10.
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