What are the Euler equations associated with the Lagrangian L(x, Y, Y' , z, z') = (y')²z – 3xz'y O 3xz – 2y" z –- 2y 2 = 0 and (y')² – 3y – 3xy = 0 O 3x2 + 2y" z + 2y' z = 0 and (y/)² – 3y – 3xy = 0 O more than one of these answers O 3xz + 2y" z + 2y' z = 0 and (y)² + 3y+ 3æy' = 0 O 3xz – 2y" z – 2y 2 = 0 and (y')² + 3y+ 3xy/ = 0 O none of these answers

1) Show that the given function y(x) = C,e*+C,e*cosx is a solution to the equation y"-3y" +4y'-2y 0 for any pair of constants (C, ,C) 2) Show that there is no pair of constants (C, ,C, for which y(x) above satisfies the following ICs y(0) = 1, y'(0) = 0, y"(0) = 0

10) Analyze all ordinary and singular points for the following DE's. Must show all work! a) 2x(x-2)'y" + 3xy' + (x-2)y=0. b) (lma)y" + y + y = 0.

+) Solve Poisson problem for a=lı bal ру u(x₁b) = f₂(x) Du= f(x,y) u(0₁9)=9₁ (9) b 0 u(a.y) = 9₂ (y) >x a u(x₁) = f(x) Figure 1. f(x,y) = sin2TX₁ f₁ = £₂=0, 9₁=9₂=0 using one dimensional eigen function expansion.