3. (Sec. 3.5) Consider the nonhomogeneous second order linear DE Ly] =y" +p(t)y' + g(t)y=g(t), where p, q, and g are continuous on an open interval I. Let Y₁ and Y₂ be two solutions of (1). (a) Show that Y₁ - Y₂ is a solution of L[y] = 0. (b) If y₁ and 2 form a set of fundamental solutions, then show that Y₁(t)-Y₂(t) = c(t) + 2/2 (t (1)

1) Consider the function z = ye ³x + 2x cos (y) a. Find the partial derivatives. b. Find the tangent plane to the surface when (x, y) = (-1,0). c. Find all the partial second derivatives.

2. Suppose that y₁ is a solution for of the homogeneous SOLDE " + p(t)y' + q(t)y = 0. (a) Explain how to find a non constant function tv(t) such that y2(t) = v(t)y₁(t) is also a solution. (b) Explain why y₁ and y₂ are linearly independent.

1. Find an interval for which the following has an unique solution: (x-2)y"+3y = x, y(3) =2,y'(3) = 1