The most popular fourth order Runge-Kutta method for the solution of $y ^ { t } = f ( x , y ) , y \left( x _ { 0 } \right) = y _ { 0 }$ is
A) $\begin{array} { l }
y _ { n + 1 } = y _ { n } + h \left( 2 k _ { 1 } + k _ { 2 } + k _ { 3 } + 2 k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \
k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 1 } / 2 \right) , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 2 } / 2 \right) , \
k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
\end{array}$
B) $\begin{array} { l }
y _ { n + 1 } = y _ { n } + h \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \
k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 1 } \right) , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 2 } \right) , \
k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
\end{array}$
C) $\begin{array} { l }
y _ { n + 1 } = y _ { n } + h \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \
k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 1 } / 2 \right) , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 2 } / 2 \right) , \
k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
\end{array}$
D) $\begin{array} { l }
y _ { n + 1 } = y _ { n } + h \left( 2 k _ { 1 } + k _ { 2 } + k _ { 3 } + 2 k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \
k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 1 } \right) , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 2 } \right) , \
k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
\end{array}$
E) $\begin{array} { l }
y _ { n + 1 } = y _ { n } + h \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \
k _ { 2 } = f \left( x _ { n } + h / 3 , y _ { n } + h k _ { 1 } / 2 \right) , k _ { 3 } = f \left( x _ { n } + 2 h / 3 , y _ { n } + h k _ { 2 } / 2 \right) , \
k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
\end{array}$

Question

The most popular fourth order Runge-Kutta method for the solution of $y ^ { t } = f ( x , y ) , y \left( x _ { 0 } \right) = y _ { 0 }$ is
A) $\begin{array} { l } 
y _ { n + 1 } = y _ { n } + h \left( 2 k _ { 1 } + k _ { 2 } + k _ { 3 } + 2 k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \
k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 1 } / 2 \right) , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 2 } / 2 \right) , \
k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
\end{array}$
B) $\begin{array} { l } 
y _ { n + 1 } = y _ { n } + h \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \
k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 1 } \right) , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 2 } \right) , \
k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
\end{array}$
C) $\begin{array} { l } 
y _ { n + 1 } = y _ { n } + h \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \
k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 1 } / 2 \right) , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 2 } / 2 \right) , \
k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
\end{array}$
D) $\begin{array} { l } 
y _ { n + 1 } = y _ { n } + h \left( 2 k _ { 1 } + k _ { 2 } + k _ { 3 } + 2 k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \
k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 1 } \right) , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 2 } \right) , \
k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
\end{array}$
E) $\begin{array} { l } 
y _ { n + 1 } = y _ { n } + h \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \
k _ { 2 } = f \left( x _ { n } + h / 3 , y _ { n } + h k _ { 1 } / 2 \right) , k _ { 3 } = f \left( x _ { n } + 2 h / 3 , y _ { n } + h k _ { 2 } / 2 \right) , \
k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
\end{array}$

Quizplus · Accepted Answer

The most popular fourth order Runge-Kutta method is described by the formula in option C, where the next value $y_{n+1}$ is calculated using a weighted average of four increments, $k_1$, $k_2$, $k_3$, and $k_4$, with specific coefficients (1, 2, 2, 1) and the increments calculated at different points within the interval using previous increments. This method is widely used due to its accuracy and efficiency for solving ordinary differential equations.

The Most Popular Fourth Order Runge-Kutta Method for the Solution yt=f(x,y),y(x0)=y0y ^ { t } = f ( x , y ) , y \left( x _ { 0 } \right) = y _ { 0 }yt=f(x,y),y(x0​)=y0​

The Most Popular Fourth Order Runge-Kutta Method for the Solution $y ^ { t } = f ( x , y ) , y \left( x _ { 0 } \right) = y _ { 0 }$