The improved Euler's formula for solving $y ^ { \prime } = f ( x , y ) , y ( \bar { x } ) = \bar { y }$ is
A) $y _ { n + 1 } = y _ { n } - f \left( x _ { n } , y _ { n } \right) , y _ { 0 } = \bar { y } , n = 0,1,2 , \ldots$
B) $y _ { n + 1 } = y _ { n } - h f \left( x _ { n } , y _ { n } \right) , y _ { 0 } = \bar { y } , n = 0,1,2 , \ldots$
C) $y _ { n + 1 } = y _ { n } + h f \left( x _ { n } , y _ { n } \right) , y _ { 0 } = \bar { y } , n = 0,1,2 , \ldots$
D) $y _ { n + 1 } = y _ { n } + \left( f \left( x _ { n } , y _ { n } \right) + f \left( x _ { n + 1 } , y _ { n + 1 } ^ { * } \right) / 2 \right) , y _ { 0 } = \bar { y } , n = 0,1,2 , \ldots \text { where } y _ { n + 1 } ^ { * }$
is predicted from Euler's formula
E) $y _ { n + 1 } = y _ { n } + h \left( f \left( x _ { n } , y _ { n } \right) + f \left( x _ { n + 1 } , y _ { n + 1 } ^ { * } \right) / 2 \right) , y _ { 0 } = \bar { y } , n = 0,1,2 , \ldots \text { where } y _ { n + 1 } ^ { * }$
is predicted from Euler's formula

Question

Quizplus · Accepted Answer

The improved Euler's method, also known as the Heun's method, involves an initial prediction of $y_{n+1}$ using Euler's formula and then correcting this prediction by averaging the slopes at the beginning and end of the interval. This is accurately described in option E, where the step size $h$ is included in the formula, and the average of the initial and predicted slopes is used to update $y_n$.

The Improved Euler's Formula for Solving y′=f(x,y),y(xˉ)=yˉy ^ { \prime } = f ( x , y ) , y ( \bar { x } ) = \bar { y }y′=f(x,y),y(xˉ)=yˉ​

The Improved Euler's Formula for Solving $y ^ { \prime } = f ( x , y ) , y ( \bar { x } ) = \bar { y }$