B2. Suppose the non-linear system dx/dt = f(x, y), dy/dt = g(x,y) has a fixed point at (x*,*). Derive the linearisation of the system about (x*,*). Under what circumstances are the solution paths of the non-linear system close to those of the linearisation? In a model of pollution, capital K and pollution P satisfy the pair of differential equations dk/dt = K(ska-1-8), dP/dt = K³ −yP where 0 0, y > 0 and ẞ>1. Find the fixed point (K*,P*) satisfying K*> 0 and P* > 0 and show it is locally stable for the non-linear system. Find an explicit expression for K(t) when K(0) = K₁ ≥0, and find its limit as t→∞.
B2. Suppose the non-linear system dx/dt = f(x, y), dy/dt = g(x,y) has a fixed point at (x*,*). Derive the linearisation of the system about (x*,*). Under what circumstances are the solution paths of the non-linear system close to those of the linearisation? In a model of pollution, capital K and pollution P satisfy the pair of differential equations dk/dt = K(ska-1-8), dP/dt = K³ −yP where 0 0, y > 0 and ẞ>1. Find the fixed point (K*,P*) satisfying K*> 0 and P* > 0 and show it is locally stable for the non-linear system. Find an explicit expression for K(t) when K(0) = K₁ ≥0, and find its limit as t→∞.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 12CR
Question
![B2.
Suppose the non-linear system
dx/dt = f(x, y), dy/dt = g(x,y)
has a fixed point at (x*,*). Derive the linearisation of the system about (x*,*).
Under what circumstances are the solution paths of the non-linear system close to those
of the linearisation?
In a model of pollution, capital K and pollution P satisfy the pair of differential equations
dk/dt = K(ska-1-8), dP/dt = K³ −yP
where 0<s<1, 0 < a < 1, &> 0, y > 0 and ẞ>1.
Find the fixed point (K*,P*) satisfying K*> 0 and P* > 0 and show it is locally stable
for the non-linear system.
Find an explicit expression for K(t) when K(0) = K₁ ≥0, and find its limit as t→∞.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67e0ae11-d999-4433-890d-8889fe301377%2F8fe9c459-24db-4d0b-beff-9f11672b85da%2Fv0mwheb_processed.png&w=3840&q=75)
Transcribed Image Text:B2.
Suppose the non-linear system
dx/dt = f(x, y), dy/dt = g(x,y)
has a fixed point at (x*,*). Derive the linearisation of the system about (x*,*).
Under what circumstances are the solution paths of the non-linear system close to those
of the linearisation?
In a model of pollution, capital K and pollution P satisfy the pair of differential equations
dk/dt = K(ska-1-8), dP/dt = K³ −yP
where 0<s<1, 0 < a < 1, &> 0, y > 0 and ẞ>1.
Find the fixed point (K*,P*) satisfying K*> 0 and P* > 0 and show it is locally stable
for the non-linear system.
Find an explicit expression for K(t) when K(0) = K₁ ≥0, and find its limit as t→∞.
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