Use the fourth-order Runge-Kutta subroutine with h = 0.25 to approximate the solution to the initial value problem below, at x = 1. Using the Taylor method of order 4, the solution to the initial value problem, at x = 1, is $\phi_T(1) = 2.125065$. Compare your approximation with the one obtained using the Taylor method. Note whether or not $\frac{\partial f}{\partial y}$ is bounded.
$\qquad y' = x + 1 - y, \quad y(0) = 3$
Let $y' = f(x, y)$. Find $\frac{\partial f}{\partial y}$ and determine whether or not it is bounded on the vertical strip $S = \{(x, y): 0 < x < 1, -\infty < y < \infty\}$. Select the correct choice below and fill in the answer box to complete your choice.
A. $\frac{\partial f}{\partial y}(x, y) = $ is not bounded.
B. $\frac{\partial f}{\partial y}(x, y) = -1$ is bounded.
Use the fourth-order Runge-Kutta subroutine with h = 0.25 to approximate the solution to the initial value problem at x = 1.
$\phi_4(1) = $ (Round to six decimal places as needed.)
