Find the non trivial solution of advection diffusion boundary value problems ut = uxx-2ux , t>=0, 0<=x<=1 u(0,t)=u(1,t)=0, t>=0

Prove the uniqueness of the diffusion problemwith Neumann boundary conditions

Minimize ƒ(x, y, z) = xy + yz subject x2 + y2 - 2 = 0 and x2 + z2 - 2 = 0.

16.Consider the autonomous system dxdt=y,dydt=x+2x3.dxdt=y,dydt=x+2x3. a.Show that the critical point (0, 0) is a saddle point. b.Sketch the trajectories for the corresponding linear system by integrating the equation for dy/dx. Show from the parametric form of the solution that the only trajectory on which x → 0, y → 0 as t → ∞ is y = −x. c.Determine the trajectories for the nonlinear system by integrating the equation for dy/dx. Sketch the trajectories for the nonlinear system that correspond to y = −x and y = x for the linear system.