91Ó°ÊÓ

Show that when the improved Euler’s method is used to approximate the solution of the initial value problem $\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{4}\mathbf{y}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}$, at$\mathbf{x}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}$ , then the approximation with step size his$\frac{\mathbf{1}}{\mathbf{3}}{\left(1+4h+8h^2\right)}^{\frac{\mathbf{1}}{\mathbf{2}\mathbf{h}}}$ .

Use the improved Euler’s method subroutine with step size h= 0.1 to approximate the solution to the initial value problem$\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{x}\mathbf{=}{\mathbf{y}}^{\mathbf{2}}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{0}$, at the points x= 1.1, 1.2, 1.3, 1.4, and 1.5. (Thus, input N= 5.) Compare these approximations with those obtained using Euler’s method (see Exercises 1.4,Problem 5, page 28).

Use the improved Euler’s method subroutine with step size h= 0.2 to approximate the solution to the initial value problem$\mathbf{y}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{x}}\mathbf{(}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}\mathbf{y}\mathbf{)}\mathbf{,}\mathbf{y}\mathbf{(}\mathbf{1}\mathbf{)}\mathbf{=}\mathbf{1}$ at the points x= 1.2, 1.4, 1.6, and 1.8. (Thus, input N= 4.) Compare these approximations with those obtained using Euler’s method (see Exercises 1.4, Problem 6, page 28).

Use the improved Euler’s method subroutine with step size h = 0.2 to approximate the solution toat the points x = 0, 0.2, 0.4, …., 2.0. Use your answers to make a rough sketch of the solution on [0, 2].

Use the fourth-order Runge–Kutta algorithm to approximate the solution to the initial value problem$\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{xy}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{1}$at x = 2. For a tolerance of, use a stopping procedure based on the absolute error.

Determine the recursive formulas for the Taylor method of order 2 for the initial value problem$\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{cos}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{)}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{Ï€}$.

Determine the recursive formulas for the Taylor method of order 2 for the initial value problem$\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{xy}\mathbf{-}{\mathbf{y}}^{\mathbf{2}}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{1}$.

Determine the recursive formulas for the Taylor method of order 4 for the initial value problem$\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{x}\mathbf{-}\mathbf{y}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$.

Determine the recursive formulas for the Taylor method of order 4 for the initial value problem $\mathbf{y}\mathbf{\text{'}}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{y}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$.

Use the Taylor methods of orders 2 and 4 with h = 0.25 to approximate the solution to the initial value problem $\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{-}\mathbf{y}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}$, at x = 1. Compare these approximations to the actual solution$\mathbf{y}\mathbf{=}\mathbf{x}\mathbf{+}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}$ evaluated at x = 1.