Solved

The Fourth Order Runge-Kutta Method for Solving Is
A) Where m1=un,k1=f(xn,yn,un)m _ { 1 } = u _ { n } , k _ { 1 } = f \left( x _ { n } , y _ { n } , u _ { n } \right)

Question 1

Multiple Choice

The fourth order Runge-Kutta method for solving y′′=f(x,y,y′) ,y(x0) =y0,y′(x0) =u0y ^ { \prime \prime } = f \left( x , y , y ^ { \prime } \right) , y \left( x _ { 0 } \right) = y _ { 0 } , y ^ { \prime } \left( x _ { 0 } \right) = u _ { 0 } is


A) yn+1=yn+h(m1−2m2+2m3−m4) /6,un+1=un+h(k1−2k2+2k3−k4) /6\begin{array} { l } y _ { n + 1 } = y _ { n } + h \left( m _ { 1 } - 2 m _ { 2 } + 2 m _ { 3 } - m _ { 4 } \right) / 6 , \\u _ { n + 1 } = u _ { n } + h \left( k _ { 1 } - 2 k _ { 2 } + 2 k _ { 3 } - k _ { 4 } \right) / 6\end{array}
where m1=un,k1=f(xn,yn,un) m _ { 1 } = u _ { n } , k _ { 1 } = f \left( x _ { n } , y _ { n } , u _ { n } \right) ,
m2=un+hk1/2,k2=f(xn+h/2,yn+hm1/2,un+hk1/2) m3=un+hk2/2,k3=f(xn+h/2,yn+hm2/2,un+hk2/2) m4=un+hk3,k4=f(xn+h,yn+hm3,un+hk3) \begin{array} { l } m _ { 2 } = u _ { n } + h k _ { 1 } / 2 , k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h m _ { 1 } / 2 , u _ { n } + h k _ { 1 } / 2 \right) \\m _ { 3 } = u _ { n } + h k _ { 2 } / 2 , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h m _ { 2 } / 2 , u _ { n } + h k _ { 2 } / 2 \right) \\m _ { 4 } = u _ { n } + h k _ { 3 } , k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h m _ { 3 } , u _ { n } + h k _ { 3 } \right) \end{array}
B) yn+1=yn−h(m1+2m2+2m3+m4) /6,un+1=un−h(k1+2k2+2k3+k4) /6\begin{array} { l } y _ { n + 1 } = y _ { n } - h \left( m _ { 1 } + 2 m _ { 2 } + 2 m _ { 3 } + m _ { 4 } \right) / 6 , \\u _ { n + 1 } = u _ { n } - h \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right) / 6\end{array}
where m1=un,k1=f(xn,yn,un) m _ { 1 } = u _ { n } , k _ { 1 } = f \left( x _ { n } , y _ { n } , u _ { n } \right) ,
m2=un+hk1/2,k2=f(xn+h/2,yn+hm1/2,un+hk1/2) m3=un+hk2/2,k3=f(xn+h/2,yn+hm2/2,un+hk2/2) m4=un+hk3,k4=f(xn+h,yn+hm3,un+hk3) \begin{array} { l } m _ { 2 } = u _ { n } + h k _ { 1 } / 2 , k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h m _ { 1 } / 2 , u _ { n } + h k _ { 1 } / 2 \right) \\m _ { 3 } = u _ { n } + h k _ { 2 } / 2 , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h m _ { 2 } / 2 , u _ { n } + h k _ { 2 } / 2 \right) \\m _ { 4 } = u _ { n } + h k _ { 3 } , k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h m _ { 3 } , u _ { n } + h k _ { 3 } \right) \end{array}
C) yn+1=yn+h(m1+2m2+2m3+m4) /6,un+1=un−h(k1+2k2+2k3+k4) /6\begin{array} { l } y _ { n + 1 } = y _ { n } + h \left( m _ { 1 } + 2 m _ { 2 } + 2 m _ { 3 } + m _ { 4 } \right) / 6 , \\u _ { n + 1 } = u _ { n } - h \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right) / 6\end{array}
where m1=un,k1=f(xn,yn,un) m _ { 1 } = u _ { n } , k _ { 1 } = f \left( x _ { n } , y _ { n } , u _ { n } \right) ,
m2=un+hk1/2,k2=f(xn+h/2,yn+hm1/2,un+hk1/2) m3=un+hk2/2,k3=f(xn+h/2,yn+hm2/2,un+hk2/2) m4=un+hk3,k4=f(xn+h,yn+hm3,un+hk3) \begin{array} { l } m _ { 2 } = u _ { n } + h k _ { 1 } / 2 , k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h m _ { 1 } / 2 , u _ { n } + h k _ { 1 } / 2 \right) \\m _ { 3 } = u _ { n } + h k _ { 2 } / 2 , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h m _ { 2 } / 2 , u _ { n } + h k _ { 2 } / 2 \right) \\m _ { 4 } = u _ { n } + h k _ { 3 } , k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h m _ { 3 } , u _ { n } + h k _ { 3 } \right) \end{array}
D) yn+1=yn−h(m1+2m2+2m3+m4) /6,un+1=un+h(k1+2k2+2k3+k4) /6\begin{array} { l } y _ { n + 1 } = y _ { n } - h \left( m _ { 1 } + 2 m _ { 2 } + 2 m _ { 3 } + m _ { 4 } \right) / 6 , \\u _ { n + 1 } = u _ { n } + h \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right) / 6\end{array}
where m1=un,k1=f(xn,yn,un) m _ { 1 } = u _ { n } , k _ { 1 } = f \left( x _ { n } , y _ { n } , u _ { n } \right) ,
m2=un+hk1/2,k2=f(xn+h/2,yn+hm1/2,un+hk1/2) m3=un+hk2/2,k3=f(xn+h/2,yn+hm2/2,un+hk2/2) m4=un+hk3,k4=f(xn+h,yn+hm3,un+hk3) \begin{array} { l } m _ { 2 } = u _ { n } + h k _ { 1 } / 2 , k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h m _ { 1 } / 2 , u _ { n } + h k _ { 1 } / 2 \right) \\m _ { 3 } = u _ { n } + h k _ { 2 } / 2 , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h m _ { 2 } / 2 , u _ { n } + h k _ { 2 } / 2 \right) \\m _ { 4 } = u _ { n } + h k _ { 3 } , k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h m _ { 3 } , u _ { n } + h k _ { 3 } \right) \end{array}
E) yn+1=yn+h(m1+2m2+2m3+m4) /6,un+1=un+h(k1+2k2+2k3+k4) /6\begin{array} { l } y _ { n + 1 } = y _ { n } + h \left( m _ { 1 } + 2 m _ { 2 } + 2 m _ { 3 } + m _ { 4 } \right) / 6 , \\u _ { n + 1 } = u _ { n } + h \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right) / 6\end{array}
where m1=un,k1=f(xn,yn,un) m _ { 1 } = u _ { n } , k _ { 1 } = f \left( x _ { n } , y _ { n } , u _ { n } \right) ,
m2=un+hk1/2,k2=f(xn+h/2,yn+hm1/2,un+hk1/2) m3=un+hk2/2,k3=f(xn+h/2,yn+hm2/2,un+hk2/2) m4=un+hk3,k4=f(xn+h,yn+hm3,un+hk3) \begin{array} { l } m _ { 2 } = u _ { n } + h k _ { 1 } / 2 , k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h m _ { 1 } / 2 , u _ { n } + h k _ { 1 } / 2 \right) \\m _ { 3 } = u _ { n } + h k _ { 2 } / 2 , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h m _ { 2 } / 2 , u _ { n } + h k _ { 2 } / 2 \right) \\m _ { 4 } = u _ { n } + h k _ { 3 } , k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h m _ { 3 } , u _ { n } + h k _ { 3 } \right) \end{array}

Correct Answer:

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