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Runge–Kutta Method (4th Order, 1st-Order ODE)

TOP/Numerical Analysis/Ordinary differential equations
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Find the solution y=f(x) of the first-order ordinary differential equation y’=F(x,y) using the 4th-order Runge–Kutta method. Starting from y0=f(x0), calculate the solution for x0≦x≦xn.
\(\normalsize y'=F(x,y)\hspace{30px} y_0=f(x_0)\rightarrow\ y=f(x)\\\)
F(x,y)
x0
Initial conditions
y0
=f(x0) Initial conditions
xn
x0≦x≦xn
[ n]Number of subdivisions
\(\normalsize Runge-Kutta\ method\\ (1)\ y'=F(x,y),\hspace{30px} y_0=f(x_0)\rightarrow\ y=f(x)\\ (2)\ y_{n+1}=y_n+{\large\frac{1}{6}}(k_1+2k_2+2k_3+k_4)+{\small O}(h^5)\\\\ \hspace{30px} k_1=hF(x_n,\ y_n)\\ \hspace{30px} k_2=hF(x_n+{\large\frac{h}{2}},\ y_n+{\large\frac{k_1}{2}})\\ \hspace{30px} k_3=hF(x_n+{\large\frac{h}{2}},\ y_n+{\large\frac{k_2}{2}})\\ \hspace{30px}k_4=hF(x_n+h,\ y_n+k_3)\\ \)

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