Show that the stationary paths of this system satisfy the Euler-Lagrange equation d²y β dx² +\w(x)y=0, y(0) = 0, (a− yλ) y(1) + ßy′(1) = 0, where X is a Lagrange multiplier. Consider the functional S[y] = ay(1)² + | " dx ẞy², y(0) = 0, with a natural boundary condition at x = 1 and subject to the constraint 1 C[y] = ry(1)² + [[" dx w(x) y² = 1, where a, ẞ and y are nonzero constants.

Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii a1=0.1 11. Consider the function f(x)=4x2(1x) a. Find any equilibrium points where f(x)=x. b. Determine the derivative at each of the equilibrium points found in part a. c. What does the theorem on the Stability of Equilibrium points tell us about each of the equilibrium points found in part a? d. Find the next four iterations of the function for the following starting values. i. a1=0.4. ii. a2=0.7 e. Describe the behavior of successive iteration found in part d. f. Discuss how the behavior found in part d relates to the results from part c.

Find y as a function of if y(1) = 3, y'(1) = -7. y = r²y" + 3xy' - 63y = 0,

11) Find a function y-f (x) that satisfies both conditions: dy = 6x-4x+5, f(-1) =3 dx dy 2 3 ==++2x-1, f (1) =1 dx x x"