Question

Using the classical fourth order Runge-Kutta method, find the approximate value of y(0.6) for the initial value problem dy/dx = sin xy ; y(0) =1 with the step size h =0.2

Answer 1

$$y_{n+1}=y_n+h(\frac {k_1}{6}+\frac{k_2}{3}+\frac{k_3}{3}+\frac {k_4}{6})$$

$$x_{n+1}=x_n+h$$

$$k_1=f(x_n;y_n)$$

$$k_2=f(x_n+\frac{h}{2}; y_n+\frac {hk_1}{2})$$

$$k_3=f(x_n+\frac{h}{3}; y_n+\frac {hk_2}{2})$$

$$k_4=f(x_n+h; y_n+hk_3)$$

Stability function:

$$\frac{z^4}{24}+\frac{z^3}{6}+\frac{z^2}{2}+z+1$$