The Euler formula for solving the system $y ^ { t } = u , u ^ { t } = f ( x , y , u ) , y \left( x _ { 0 } \right) = y _ { 0 } , u \left( x _ { 0 } \right) = u _ { 0 }$ is
A) $y _ { n + 1 } = y _ { n } + h u _ { n } , u _ { n + 1 } = u _ { n } + h f \left( x _ { n } , y _ { n } , u _ { n } \right)$
B) $y _ { n + 1 } = y _ { n } - h u _ { n } , u _ { n + 1 } = u _ { n } + h f \left( x _ { n } , y _ { n } , u _ { n } \right)$
C) $y _ { n + 1 } = y _ { n } + h u _ { n } , u _ { n + 1 } = u _ { n } - h f \left( x _ { n } , y _ { n } , u _ { n } \right)$
D) $y _ { n + 1 } = y _ { n } + h f \left( x _ { n } , y _ { n } , u _ { n } \right) , u _ { n + 1 } = u _ { n } + h u _ { n }$
E) none of the above

Question

Quizplus · Accepted Answer

The Euler method for solving differential equations updates the next value of the dependent variable based on the current slope. Here, $y_{n+1} = y_n + h u_n$ updates $y$ based on the current value of $u$, and $u_{n+1} = u_n + h f(x_n, y_n, u_n)$ updates $u$ based on the current derivative $f(x, y, u)$, which matches the system given.

The Euler Formula for Solving the System yt=u,ut=f(x,y,u),y(x0)=y0,u(x0)=u0y ^ { t } = u , u ^ { t } = f ( x , y , u ) , y \left( x _ { 0 } \right) = y _ { 0 } , u \left( x _ { 0 } \right) = u _ { 0 }yt=u,ut=f(x,y,u),y(x0​)=y0​,u(x0​)=u0​

The Euler Formula for Solving the System $y ^ { t } = u , u ^ { t } = f ( x , y , u ) , y \left( x _ { 0 } \right) = y _ { 0 } , u \left( x _ { 0 } \right) = u _ { 0 }$