Kindly don't copy solution from (ch**g) website, solve it on your own

1. Given a vector field with a critical point at the origin and a function L, determine if L is a first integral and check whether L is a Lyapunov function near the origin. (a) x = y+ x2, y = -x+ y', L(x, y) =x2+ y2; (b) ic = 2x - 4x2y3, y = -3y + 6xy4, L(x, y) = x3y2; (c) x = sin y, y = - sin x, L(x, y) = cos x + cosy; (d) x = sing + 0.1 sinx, y = - sin x + 0.2 siny, L(x, y) = cos x + cosy; (e) x = sin y + 0.1 sinx, y = - sin x - 0.1 siny, L(x, y) = cosx + cosy. Sketch phase portraits for the last three systems near the origin and explain what qualitative properties of the pictures relate to different answers in these parts. 2. Find a first integral of the following second-order ODE: x = cosx, then check whether it is a Lyapunov function for * = cos x-0.1x near the point (x, x) = (#, 0)