Explanation Formula used: The Trapezoidal scheme is, y n + 1 = y n + h 2 [ f ( x n , y n ) + f ( x n + 1 , y n + 1 ) ] , n = 0 , 1 , 2 , ... Calculation: Consider the given initial value problem. y ' = y , y ( 0 ) = 1 Consider that, f ( x , y ) = y ' = y Start with x 0 = 0 and y 0 = 1 Consider that step size is h so, the number of approximation required for x n = 1 is 1 h . Consider the i th step of Trapezoidal scheme, y i + 1 = y i + h 2 [ f ( x i , y i ) + f ( x i + 1 , y i + 1 ) ] = y i + h 2 [ f ( x i , y i ) + f ( x i + h , y i + h f ( x i , y i ) ) ] = y i + h 2 [ y i + f ( x i + h , y i + h y i ) ] = y i + h 2 [ 2 y i + h y i ] Simplify the above equation. y i + 1 = y i + ( 1 + h + h 2 2 ) Multiply by 2 − h on both sides. y i + 1 ( 2 − h ) = y i + ( 1 + h + h 2 2 ) ( 2 − h ) = ( 2 + 2 h + h 2 − h − h 2 − h 2 2 ) y i = ( 2 + h − h 3 2 ) y i = ( 2 + h ) y i − h 3 2 y i Ignore the term h 3 2 as it is very small for numerical considerations. y i + 1 ( 2 − h ) = ( 2 + h ) y i So, consider above equation, y n = ( 2 + h ) ( 2 − h ) y n − 1 And y n = ( ( 2 + h ) ( 2 − h ) ) n y 0 Substitute 1 h for n in above relation. y n = ( ( 2 + h ) ( 2 − h ) ) 1 h y 0 = ( ( 1 + h 2 ) ( 1 − h 2 ) ) 1 h y 0 So, at x = 1 , y 0 is 0 , we have, y ( 1 ) = ( ( 1 + h 2 ) ( 1 − h 2 ) ) 1 h …(1) Substitute 1for h in equation ( 1 )

Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

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Author: R. Nagle, Edward Saff

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Chapter 3.6, Problem 3E

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The solution of initial value problem y'=y,y(0)=1 at x=1 with step size h is (1+h21−h2)1h using trapezoidal scheme.

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Chapter 3.6, Problem 2E

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Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

Ch. 3.2 - A brine solution of salt flows at a constant rate...Ch. 3.2 - Prob. 2E Ch. 3.2 - Prob. 3E Ch. 3.2 - A brine solution of salt flows at a constant rate...Ch. 3.2 - A swimming pool whose volume is 10,000gal contains...Ch. 3.2 - The air in a small room 12ft by 8ft by 8ft is 3...Ch. 3.2 - Beginning at time t=0, fresh water is pumped at...Ch. 3.2 - A tank initially contains S0lb of salt dissolved...Ch. 3.2 - In 1990 the Department of Natural Resources...Ch. 3.2 - Prob. 10E

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The solution of initial value problem y ' = y , y ( 0 ) = 1 at x = 1 with step size h is ( 1 + h 2 1 − h 2 ) 1 h using trapezoidal scheme.

Solution Summary: The author explains that the solution to initial value problem y'=y,y

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The solution of initial value problem y'=y,y(0)=1 at x=1 with step size h is (1+h21−h2)1h using trapezoidal scheme.

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The solution of initial value problem y ' = y , y ( 0 ) = 1 at x = 1 with step size h is ( 1 + h 2 1 − h 2 ) 1 h using trapezoidal scheme.

Solution Summary: The author explains that the solution to initial value problem y'=y,y

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To show: The solution of initial value problem y'=y,y(0)=1 at x=1 with step size h is (1+h21−h2)1h using trapezoidal scheme.

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Chapter 3 Solutions

To show:

The solution of initial value problem y'=y,y(0)=1 at x=1 with step size h is (1+h21−h2)1h using trapezoidal scheme.