# Runge-Kutta method (2nd-order,2nd-derivative) Calculator

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

 The initial condition is y0=f(x0), y'0=p0=f'(x0) and the root x is calculated within the range of from x0 to xn. $\normal y''=F(x,y,y')\hspace{30} y_0=f(x_0),\ y'_0=f'(x_0) \rightarrow\ y=f(x)\\$
 F(x,y,p(=y')) x0 initial condition y0 = f(x0) y'0=p0 = f'(x0) xn x0≦x≦xn [ partition n 10 20 50 100 200 500  ] 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit
 $\normal Runge-Kutta\ method\\[10pt](1)\ y''=F(x,y,y')\hspace{30} p=y',\ y_0=f(x_0),\ y'_0=f'(x_0) \rightarrow\ y=f(x)\\(2)\ p_{n+1}=p_n+j_2+{\small O}(h^3)\\\vspace{10}\\\hspace{25} y_{n+1}=y_n+k_2+{\small O}(h^3)\\\vspace{10}\\\hspace{25} j_1=hF(x_n,\ y_n,\ p_n)\\\hspace{25} k_1=h \cdot p_n \\\hspace{25} j_2=hF(x_n+{\large\frac{h}{2}},\ y_n+{\large\frac{k_1}{2}},\ p_n+{\large\frac{j_1}{2}})\\\hspace{25} k_2=h ( p_n + {\large\frac{j_1}{2}} )\\$
Runge-Kutta method (2nd-order,2nd-derivative)
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