Chapter 3.6, Problem 8E

Use the improved Euler’s method subroutine with step size $h=0.2$ to approximate the solution to the initial value problem

$y^{\prime }=\frac{1}{x}\left(y^2+y\right)$, $y\left(1\right)=1$,

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, page28).

6. Use the Euler’s method with step size $h=0.2$ to approximate the solution to the initial value problem

$y^{\prime }=\frac{1}{x}\left(y^2+y\right)$, $y\left(1\right)=1$

at the points $x=1.2$, $1.4$, $1.6$, and $1.8$.