When working on these problems, it is recommended to use Matlab or Java since these problems require many calculations. Show your work and explain your answers.

2. [5 pts] Use the shooting method with Taylor's method of order 3 on the following BVP. d x = x' + sin(t) dt2 x(0) = 1, x(0.5) = 2 Let z1 = 1.1 be the first guess, and calculate 0(zi) using Taylor's method of order 3 and 21. Write your answer in the blank below for 0(21). Let 22 = 1.4 be the second guess, and calculate 0(22) using Taylor's method of order 3 and 22. Write your answer in the blank below for $(22). Use linear interpolation to calculate 23, and calculate (23) using Taylor's method of order 3 and zz. Write your answer for 23 and 0(23) in their blanks below. Use Taylor's method of order 3 and 23 to approximate x(0.25). Write your answer in the blank below for x(0.25). (a) (21) = (b) 0(22) = (c) 23 = (d) (23) = (e) x(0.25) ~ 2. [5 pts] Use the shooting method with Taylor's method of order 3 on the following BVP. d x = x' + sin(t) dt2 x(0) = 1, x(0.5) = 2 Let z1 = 1.1 be the first guess, and calculate 0(zi) using Taylor's method of order 3 and 21. Write your answer in the blank below for 0(21). Let 22 = 1.4 be the second guess, and calculate 0(22) using Taylor's method of order 3 and 22. Write your answer in the blank below for $(22). Use linear interpolation to calculate 23, and calculate (23) using Taylor's method of order 3 and zz. Write your answer for 23 and 0(23) in their blanks below. Use Taylor's method of order 3 and 23 to approximate x(0.25). Write your answer in the blank below for x(0.25). (a) (21) = (b) 0(22) = (c) 23 = (d) (23) = (e) x(0.25) ~