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→∞.