Chapter 2.6, 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 five decimal places by the use of Runge Kutta’s method.

Summary Introduction:

Purpose will use Runge Kutta’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 $y_{i+1}=y_i+k\cdot h$ and the value of $k$ can be calculated as $k=\frac{1}{6}\left(k_1+2k_2+2k_3+k_4\right)$ where the value of $k_1=f\left(x_i,y_i\right)$ , $k_2=f\left(x_i+\frac{1}{2}h,y_i+\frac{1}{2}h{k}_1\right)$ , $k_3=f\left(x_i+\frac{1}{2}h,y_i+\frac{1}{2}h{k}_2\right)$ and $k_4=f\left(x_{i+1},y_i+h{k}_3\right)$ .