Consider the functional S[y] = ay(1)² + [* dx ßy², y(0) = 0, with a natural boundary condition at x C{y] = Yy(1)² + [" dx w(x) y² = 1, where a, ẞ and y are nonzero constants. = 1 and subject to the constraint Show that the stationary paths of this system satisfy the Euler-Lagrange equation d²y β +\w(x) y = 0, y(0) = 0, (a− yλ) y(1) + ßy′(1) = 0, dx2 where is a Lagrange multiplier.

The first-order DE is a linear equation if it has the following form: dy a(1)+a,(»)x= g(x)“ А dx B dx a(x)+a,(x)y = g) dy В dy •.. © a, (x)+ ) + a,(x)y = g(x) C dx

Find the maximum and minimum of f(x,y) = yx2 + y(y - 1)2 subject to x2 + y2 =1.

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→∞.

Find the extremum of f(x,y) subject to the given constraint, and state whether it is a maximum or a minimum. f(x,y) = 3x² + 3y²; 2x + 3y = 39 Find the Lagrange function F(x,y,^). F(x,y,^)=-^ Find the partial derivatives Fx, Fy, and F₁. Fy = There is a value of located at (x, y) = | -0. (Type an integer or a fraction. Type an ordered pair, using integers or fractions.)