Consider the ODE - y" − k²y = f(x) 0 ≤ x ≤1 k > 0 with boundary conditions (a) y(0) = 0 y(1) = 0 Compute the Green's function G(x, x') for this ODE. That is, solve the boundary value problem (b) d²G − k²G(x, x') = d(x − x') dx² with boundary conditions G(x = 0, x') = 0 G(x = 1, x') = 0 and where 6(x) is the delta function. Use the Green's function developed in part (a) to solve the inhomo- geneous boundary value problem y" - k²y = 1 0 ≤ x ≤ 1 k > 0 with boundary conditions y(0) = 0 y(1) = 0 You may leave your answer in terms of integrals.

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.

YOUR TURN Suppose x are y are both functions of t and x3+2xy+y2=1. If x=1, y=2, and dxdt=6, then find dydt.

Use Eulers method to approximate the solution of dydtx2y2=1, with y(0)=2, for [0,1]. Use h=0.2.