Explanation Given: The initial value problem is, y ′′ ′ + y ′′ + y 2 = t y ( 0 ) = 1 y ′ ( 0 ) = 0 y ′′ ( 0 ) = 1 and step-size is, h = 0.05 . Approach: Convert the given initial value problem into initial value problem for a system in normal form. Assume, x 1 ( t ) = y ( t ) x 2 ( t ) = y ′ ( t ) x 3 ( t ) = y ′′ ( t ) ( 1 ) Compare the normal form with- x ′ 1 ( t ) = f 1 ( t , x 1 , x 2 , … , x n ) , x ′ 2 ( t ) = f 2 ( t , x 1 , x 2 , … , x n ) , ⋮ x ′ n ( t ) = f n ( t , x 1 , x 2 , … , x n ) ( 2 ) To compute an approximation for the solution to the initial value problem- x ′ i = f i ( t , x 1 , … , x n ) x i ( t 0 ) = a i i = 1 , 2 , … , n for t 0 ≤ t ≤ c , use a programmed version of the vectorized classical fourth-order Runge-Kutta algorithm with step-size h , number of steps N - t = t + h x i = x i + 1 6 ( k i , 1 + 2 k i , 2 + 2 k i , 3 + k i , 4 ) i = 1 , . … , n n = 0 , 1 , 2 , … Here, h = c − t 0 N t = t 0 x 1 = a 1 , … , x n = a n For i = 1 , … , n , k i , 1 = h f i ( t , x 1 , … , x n ) k i , 2 = h f i ( t + h 2 , x 1 + 1 2 k 1 , 1 , … , x n + 1 2 k n , 1 ) k i , 3 = h f i ( t + h 2 , x 1 + 1 2 k 1 , 2 , … , x n + 1 2 k n , 2 ) k i , 4 = h f i ( t + h , x 1 + k 1 , 3 , … , x n + k n , 3 ) Calculation: Differentiate system of equation ( 1 ) with respect to t . x ′ 1 ( t ) = x 2 ( t ) x ′ 2 ( t ) = x 3 ( t ) x ′ 3 ( t ) = y ′′′ ( t ) ( 3 ) Do substitution from equation ( 3 ) in the given initial value problem

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To approximate: The solution to the given initial value problem at t = 1 .

Solution Summary: The author explains how to approximate the solution to the given initial value problem at t=1.

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To approximate:

The solution to the given initial value problem at t=1.

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Chapter 5 Solutions

To approximate: The solution to the given initial value problem at t = 1 .

Solution Summary: The author explains how to approximate the solution to the given initial value problem at t=1.

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To approximate: The solution to the given initial value problem at t=1.

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Chapter 5 Solutions

To approximate:

The solution to the given initial value problem at t=1.