Find the vertex of the paraboloid z=x2+4y2−xy+6x−5y+1

ts) Find two linearly independent solutions of 2x²y" - xy + (2x + 1)y=0, x > 0 of the form Y₁ = x¹(1+ a₁ + a₂c² + a3x³ + ...) Y₂ = x²(1 + b₁c + b₂x² + b3x³ + ...) where T1 T2. Enter T1 T2 a1 = a₂ = a3 = b₁ b₂ TE b3 N - || || || || =

Find two linearly independent solutions of y″+2xy=0 of the form and enter the first few coefficients shown in the photo

= 12. (a) Find the unique quadratic polynomial y ax²+bx+c that passes through the three points (1, 1), (3,5), and (-2, 0) in the x-y plane. (Hint: this will involve a linear system where a, b, and c are the unknowns.) (b) How many data points (x, y) should be needed to uniquely determine the coefficients of an nth degree polynomial anx" +an-1² -1xn-1+...+ a₁x+ao? Explain your answer in terms of linear systems.

Find an orthogonal change of variables, x = Py, that transforms the quadratic form q(x) = x² + x²+2x1x2 − 2x2x3 into a quadratic form no cross-product form. Write the new quadratic form.