# Runge-Kutta method

## Calculates the solution y=f(x) of the linear ordinary differential equation y'=F(x,y) using Runge-Kutta method.

 The initial condition is y0=f(x0), and the root x is calculated within the range of from x0 to xn. $\normal \\\vspace{10}y'=F(x,y)\hspace{30} y_0=f(x_0)\rightarrow\ y=f(x)\\$
F(x,y)
x0
 initial condition
y0
 = f(x0)
xn
 x0≦x≦xn
 [ partition n 10 20 50 100 200 500 ]

x y=f(x)

 $\normal Runge-Kutta\ method\\[10pt](1)\ y'=F(x,y),\hspace{30} y_0=f(x_0)\rightarrow\ y=f(x)\\(2)\ y_{n+1}=y_n+{\large\frac{1}{6}}(k_1+2k_2+2k_3+k_4)+{\small O}(h^5)\\\vspace{10}\hspace{30} k_1=hF(x_n,\ y_n)\\\hspace{30} k_2=hF(x_n+{\large\frac{h}{2}},\ y_n+{\large\frac{k_1}{2}})\\\hspace{30} k_3=hF(x_n+{\large\frac{h}{2}},\ y_n+{\large\frac{k_2}{2}})\\\hspace{30}k_4=hF(x_n+h,\ y_n+k_3)\\$

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