Chapter 2.5, Problem 9P

Program Plan Intro

Program Description: Purpose ofproblem is to construct a table for the approximation solution and the actual solution of $y^{\prime }=\frac{1}{4}\left(1+y^2\right)$ up to four decimal places.

Summary Introduction:

Purpose will use Euler’s method to construct the table of the approximation solution and the actual solution $y=y\left(x\right)$ is the solution of a differential equation $\frac{dy}{dx}=f\left(x,y\right)$ with $y\left(x_0\right)=y_0$ where approximate value can be calculated with the formula $u_{i+1}=y_i+f\left(x_i,y_i\right)h$ and the improved values can be calculated with the formula $y_{i+1}=y_i+\frac{1}{2}\left[f\left(x_i,y_i\right)+f\left(x_{i+1},y_{i+1}\right)\right]h$ for an interval $\left[x_0,x_n\right]$ , $x_{i+1}=x_i+h$ .