Exam 9: Numerical Solutions of Ordinary Differential Equations

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

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

The solution of $y ^ { \prime } = x + y , y ( 0 ) = 1 \text { for } y ( 0.2 )$ , using the improved Euler's method with $h = 0.1$ is

Euler's method is what type of Runge-Kutta method?

Using the Adams-Bashforth-Moulton method from the previous three problems, the solution of $y ^ { \prime } = y , y ( 0 ) = 1$ for $y ( 0.4 )$ with $h = 0.1$ is

Using the Adams-Bashforth method from the previous problem, and using the values $y _ { 0 } = 1 , y _ { 1 } = 1.1052 , y _ { 2 } = 1.2214 , y _ { 3 } = 1.3499$ the solution $y ^ { \prime } = y , y ( 0 ) = 1$ for $y _ { n + 1 } ^ { * } = y ( 0.4 )$ with $h = 0.1$ is

Using the method from the previous problem, the solution of $y ^ { \prime } = y , y ( 0 ) = 1 \text { for } y ( 0.2 )$ with $h = 0.2$ is

The standard central difference approximation of $y ^ { \prime \prime } ( x )$ is

The improved Euler's method is what type of Runge-Kutta method?

The Adams-Bashforth formula for finding the solution of $y ^ { t } = f ( x , y ) , y \left( x _ { 0 } \right) = y _ { 0 }$ is

Which of the following are second order Runge-Kutta methods for the solution of $y ^ { t } = f ( x , y ) , y \left( x _ { 0 } \right) = y _ { 0 }$ ? Select all that apply.

In the previous problem, the local truncation error in $y _ { n + 1 }$ is

Using Euler's method on the previous problem and using a value of $h = 0.1$ , the solution for $y ( 0.2 )$ is

When entering the number $1 / 3$ into a three digit base ten calculator, the round-off error is

The local truncation error for the improved Euler's method is

The improved Euler's formula for solving $y ^ { \prime } = f ( x , y ) , y ( \bar { x } ) = \bar { y }$ is

When entering the number $1 / 3$ into a three digit base ten calculator, the actual value entered is

Using the value of $y _ { n + 1 } ^ { * }$ from the previous problem, the Adams-Moulton corrector value for the solution of $y ^ { t } = f ( x , y ) , y \left( x _ { 0 } \right) = y _ { 0 }$ is

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

When entering the number $1 / 7$ into a three digit base ten calculator, the round-off error is