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

3. Verify that L(x, y) = x2 + y is a Lyapunov function for the equation x = y + 23(1 - 2 cosy), y = -x - y'(1 - cos x) near the origin. Use this function to find a specific domain that is surely included in the attracting basin of the origin, i.e. all solutions originating from this domain tend to zero. The domain does not need to be the largest possible. 4. Rewrite the following differential equations in polar coordinates. Use the result to describe the qualitative behaviour of the solutions of these ODEs. Sketch the phase portraits of these ODEs and compare them to the predicted qualitative behaviour. (a) x = 2y, y = -2x; (b) i = 0.1x - y - x(x2 + y?)2, y = x + 0.ly -y(x2 + y2)2. 5. Find two vector fields (x, y) = (Pi(x, y), Qi(x, y)) and (x, y) = (P2(x, y), Q2(x, y)) and two functions Fi(x, y), F2(x, y) such that each function is a first integral of the corresponding vector field but their sum is not the first integral of the vector field (x, y) = (Pi(x, y) + P2(x, y), Q1(x, y) + Q2(x, y))