Explanation Given: The differential equation is y ′ = 2 x − 3 y + 1 . Concept: The value of y 1 , y 2 and y 3 is calculated using the RK4 method and with the initial values x 0 and y 0 . The general formulas used in the RK4 method are given below. k 1 = h F ( x n , y n ) k 2 = h F ( x n + h 2 , y n + k 1 2 ) k 3 = h F ( x n + h 2 , y n + k 2 2 ) k 4 = h F ( x n + h , y n + k 3 ) And y n + 1 = y n + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) The value y 4 is calculated using the Adams-Bashforth-Moulton method using the following formulas. y ′ n = f ( x n , y n ) y ′ n − 1 = f ( x n − 1 , y n − 1 ) y ′ n − 2 = f ( x n − 2 , y n − 2 ) y ′ n − 3 = f ( x n − 3 , y n − 3 ) And, y 4 ∗ = y 3 + 0.2 24 ( 55 y ′ 3 − 59 y ′ 2 + 37 y ′ 1 − 9 y ′ 0 ) , y ′ 4 = f ( x 4 , y 4 ∗ ) , y 4 = y 3 + 0.2 24 ( 9 y ′ 4 + 19 y ′ 3 − 5 y ′ 2 + y ′ 1 ) Calculation: The differential equation can be written as given below. y ' = F ( x , y ) F ( x , y ) = 2 x − 3 y + 1 The boundary conditions are given below. y 0 = f ( x 0 ) x 0 = 0 y 0 = 1 h = 0.2 For n = 0 , substitute the value of x 0 , y 0 and h in the following equations to get the values of k 1 , k 2 , k 3 and k 4 . Substitute 0 for x , 1 for y and 0.2 for h in equation k 1 = h F ( x 0 , y 0 ) to obtain the value of k 1 . k 1 = h F ( x 0 , y 0 ) = 0.2 × ( 2 ( 0 ) − 3 ( 1 ) + 1 ) = 0.2 × ( − 2 ) = − 0.4 Therefore, the value of k 1 is − 0.4 . Substitute 0 for x , − 0.4 for k 1 and 0.2 for h in equation k 2 = h F ( x 0 + h 2 , y 0 + k 1 2 ) to obtain the value of k 2 . k 2 = h F ( x 0 + h 2 , y 0 + k 1 2 ) = 0.2 × F ( 0 + 0.2 2 , 1 + − ( 0.4 ) 2 ) = 0.2 × F ( 0.1 , 0.8 ) = − 0.24 Therefore, the value of k 2 is − 0.24 . Substitute 0 for x , − 0.24 for k 2 and 0.2 for h in equation k 3 = h F ( x 0 + h 2 , y 0 + k 2 2 ) to obtain the value of k 3 . k 3 = h F ( x 0 + h 2 , y 0 + k 2 2 ) = 0.2 × F ( 0 + 0.2 2 , 1 + − ( 0.24 ) 2 ) = 0.2 × F ( 0.1 , 0.88 ) = − 0.288 Therefore, the value of k 3 is − 0.288 . Substitute 0 for x , − 0.288 for k 3 and 0.2 for h in equation k 4 = h F ( x 0 + h , y 0 + k 3 ) to obtain the value of k 4 . k 4 = h F ( x 0 + h , y 0 + k 3 ) = 0.2 × F ( 0 + 0.2 , 1 + ( − 0.288 ) ) = 0.2 × F ( 0.2 , 0.712 ) = − 0.1472 Therefore, the value of k 4 is − 0.1472 . Now for n = 0 , substitute − 0.4 for k 1 , − 0.24 for k 2 , − 0.288 for k 3 and − 0.1472 for k 4 in equation y n + 1 = y n + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) . y 1 = y 0 + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) = 1 + 1 6 ( ( − 0.4 ) + 2 ( − 0.24 ) + 2 ( − 0.288 ) + ( − 0.1472 ) ) = 1 + 1 6 × − ( 0.4 + 0.48 + 0.576 + 0.1472 ) = 0.7328 Therefore, the value of y 1 is 0.7328 . For n = 1 , substitute 0 for x 0 and 0.2 for h in equation x 1 = x 0 + h to obtain the value of x 1 as follows: x 1 = x 0 + h = 0 + 0.2 = 0.2 Therefore, the value of x 1 is 0.2 . Substitute 0.2 for x 1 , 0.7328 for y 1 and 0.2 for h in equation k 1 = h F ( x 1 , y 1 ) to obtain the value of k 1 . k 1 = h F ( x 1 , y 1 ) = 0.2 × ( 2 ( 0.2 ) − 3 ( 0.7328 ) + 1 ) = 0.2 × ( − 0.7984 ) = − 0.15968 Therefore, the value of k 1 is − 0.15968 . Substitute 0.2 for x 1 , 0.7328 for y 1 , 0.2 for h and − 0.15968 for k 1 in equation k 2 = h F ( x 1 + h 2 , y 1 + k 1 2 ) to obtain the value of k 2 . k 2 = h F ( x 1 + h 2 , y 1 + k 1 2 ) = 0.2 × F ( 0.2 + 0.2 2 , 0.7328 + − ( 0.15968 ) 2 ) = 0.2 × F ( 0.3 , 0.6529 ) = − 0.07174 Therefore, the value of k 2 is − 0.07174 . Substitute 0.2 for x 1 , 0.7328 for y 1 , 0.2 for h and − 0.07174 for k 2 in equation k 3 = h F ( x 1 + h 2 , y 1 + k 2 2 ) to obtain the value of k 3 . k 3 = h F ( x 1 + h 2 , y 1 + k 2 2 ) = 0.2 × F ( 0.2 + 0.2 2 , 0.7328 + − 0.07174 2 ) = 0.2 × F ( 0.3 , 0.69693 ) = − 0.098158 Therefore, the value of k 3 is − 0.098158 . Substitute 0.2 for x 1 , 0.7328 for y 1 , 0.2 for h and − 0.098158 for k 3 in equation k 4 = h F ( x 1 + h , y 1 + k 3 ) to obtain the value of k 4 . k 4 = h F ( x 1 + h , y 1 + k 3 ) = 0

Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th

11th Edition

ISBN: 9781305965737

Author: Dennis G. Zill

Publisher: Brooks Cole

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Chapter 9.3, Problem 3E

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The approximation of the indicated value y(0.8) using the Adams-Bashforth-Moulton method and use RK4 method to compute y1,y2 and y3.

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Student Solutions Manual For Zill's A First Course In Differential Equations With Modeling Applications, 11th

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The approximation of the indicated value y ( 0.8 ) using the Adams-Bashforth-Moulton method and use RK4 method to compute y 1 , y 2 and y 3 .

Solution Summary: The author describes the Adams-Bashforth-Moulton method and RK4 method to compute the indicated value y(0.8).

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The approximation of the indicated value y(0.8) using the Adams-Bashforth-Moulton method and use RK4 method to compute y1,y2 and y3.

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