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

question 19

Multiple Choice

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

• A) \(\begin{array} { l }
• y _ { n + 1 } = y _ { n } + h \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \\
• k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 1 } / 2 \right) , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 2 } / 2 \right) , \\
• k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
• \end{array}\)
• B) \(\begin{array} { l }
• y _ { n + 1 } = y _ { n } + h \left( 2 k _ { 1 } + k _ { 2 } + k _ { 3 } + 2 k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \\
• k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 1 } \right) , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 2 } \right) , \\
• k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
• \end{array}\)
• C) \(\begin{array} { l }
• y _ { n + 1 } = y _ { n } + h \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \\
• k _ { 2 } = f \left( x _ { n } + h / 3 , y _ { n } + h k _ { 1 } / 2 \right) , k _ { 3 } = f \left( x _ { n } + 2 h / 3 , y _ { n } + h k _ { 2 } / 2 \right) , \\
• k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
• \end{array}\)
• D) \(\begin{array} { l }
• y _ { n + 1 } = y _ { n } + h \left( 2 k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + 2 k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \\
• k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 1 } / 2 \right) , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 2 } / 2 \right) , \\
• k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
• \end{array}\)
• E) \(\begin{array} { l }
• y _ { n + 1 } = y _ { n } + h \left( k _ { 1 } + 2 k _ { 2 } + 2 k _ { 3 } + k _ { 4 } \right) / 6 , \text { where } k _ { 1 } = f \left( x _ { n } , y _ { n } \right) , \\
• k _ { 2 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 1 } \right) , k _ { 3 } = f \left( x _ { n } + h / 2 , y _ { n } + h k _ { 2 } \right) , \\
• k _ { 4 } = f \left( x _ { n } + h , y _ { n } + h k _ { 3 } \right)
• \end{array}\)

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Definitions:

Mean Number

The arithmetic average of a set of values, calculated by dividing the sum of all values by the number of values.

Poisson Probability

The probability of a given number of events happening in a fixed interval of time or space when these events occur with a known constant mean rate and independently of the time since the last event.

Binomial Probabilities

The probabilities of achieving a specific number of successes in a fixed number of Bernoulli trials, characterized by only two possible outcomes.

Poisson Random Variable

A variable that follows the Poisson distribution, mainly used to model the number of times an event happens in a fixed interval of time or space.