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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→∞.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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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→∞.
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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