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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 }

Question 19

Multiple Choice

The most popular fourth order Runge-Kutta method for the solution of yt=f(x,y) ,y(x0) =y0y ^ { t } = f ( x , y ) , y \left( x _ { 0 } \right) = y _ { 0 } is


A) yn+1=yn+h(k1+2k2+2k3+k4) /6, where k1=f(xn,yn) ,k2=f(xn+h/2,yn+hk1/2) ,k3=f(xn+h/2,yn+hk2/2) ,k4=f(xn+h,yn+hk3) \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}
B) yn+1=yn+h(2k1+k2+k3+2k4) /6, where k1=f(xn,yn) ,k2=f(xn+h/2,yn+hk1) ,k3=f(xn+h/2,yn+hk2) ,k4=f(xn+h,yn+hk3) \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}
C) yn+1=yn+h(k1+2k2+2k3+k4) /6, where k1=f(xn,yn) ,k2=f(xn+h/3,yn+hk1/2) ,k3=f(xn+2h/3,yn+hk2/2) ,k4=f(xn+h,yn+hk3) \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}
D) yn+1=yn+h(2k1+2k2+2k3+2k4) /6, where k1=f(xn,yn) ,k2=f(xn+h/2,yn+hk1/2) ,k3=f(xn+h/2,yn+hk2/2) ,k4=f(xn+h,yn+hk3) \begin{array} { l } y _ { n + 1 } = y _ { n } + h \left( 2 k _ { 1 } + 2 k _ { 2 } + 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}
E) yn+1=yn+h(k1+2k2+2k3+k4) /6, where k1=f(xn,yn) ,k2=f(xn+h/2,yn+hk1) ,k3=f(xn+h/2,yn+hk2) ,k4=f(xn+h,yn+hk3) \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}

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