Explanation Given: The given differential equation is, y ′ = x + 3 cos ( x y ) f ( x , y ) = x + 3 cos ( x y ) ... ( 1 ) The step size is, h = 0.2 Initial condition is, x 0 = 0 y 0 = 0 Domain of the function is [ 0 , 2 ] . Formula: The improved Euler method is given by, y n + 1 = y n + h 2 [ f ( x n , y n ) + f ( x n + h , y n + h f ( x n , y n ) ) ] ... ( 2 ) Here, n = 0 , 1 , 2 , ... Calculation: Substitute x n + 3 cos ( x n y n ) for f in equation ( 2 ) y n + 1 = y n + h 2 [ ( x n + 3 cos ( x n y n ) ) + ( x n + h + 3 cos ( ( x n + h ) ( y n + h x n + 3 h cos ( x n y n ) ) ) ) ] = y n + h 2 [ 2 x n + h + 3 cos ( x n y n ) + 3 cos ( ( x n + h ) ( y n + h x n + 3 h cos ( x n y n ) ) ) ] ... ( 3 ) Substitute 0 for n in equation ( 3 ) . y 1 = y 0 + h 2 [ 2 x 0 + h + 3 cos ( x 0 y 0 ) + 3 cos ( ( x 0 + h ) ( y 0 + h x 0 + 3 h cos ( x 0 y 0 ) ) ) ] … ( 4 ) Substitute 0.2 for h , 0 for x 0 and 0 for y 0 in equation ( 4 ) . y 1 = 0 + 0.2 2 [ 2 ( 0 ) + 0.2 + 3 cos ( 0 ⋅ 0 ) + 3 cos ( ( 0 + 0.2 ) ( 0 + 0.2 ⋅ 0 + 3 ( 0.2 ) cos ( 0 ⋅ 0 ) ) ) ] = 0.1 [ 0.2 + 3 ( 1 ) + 3 cos ( 0.2 ( 0.6 ⋅ 1 ) ) ] = 0.1 [ 3.2 + 3 cos ( 0.12 ) ] = 0.1 [ 3.2 + 3 ( 0.99280 ) ] And, y 1 = 0.1 [ 3.2 + 2.9784 ] = 0.1 [ 6.1784 ] = 0.61784 Substitute 1 for n in equation ( 3 ) . y 2 = y 1 + h 2 [ 2 x 1 + h + 3 cos ( x 1 y 1 ) + 3 cos ( ( x 1 + h ) ( y 1 + h x 1 + 3 h cos ( x 1 y 1 ) ) ) ] ... ( 5 ) Substitute 0.2 for h , 0.2 for x 1 and 0.61784 for y 1 in equation ( 5 ) . y 2 = 0.61784 + 0.2 2 [ 2 ( 0.2 ) + 0.2 + 3 cos ( 0.2 ( 0.61784 ) ) + 3 cos ( ( 0.2 + 0.2 ) ( 0.61784 + ( 0.2 ) ( 0.2 ) + 3 ( 0.2 ) cos ( ( 0.2 ) 0.61784 ) ) ) ] = 0.61784 + 0.1 [ 0.6 + 3 cos ( 0.123568 ) + 3 cos ( 0.4 ( 0.61784 + 0.04 + 0.6 cos ( 0.123568 ) ) ) ] = 0.61784 + 0.1 [ 0.6 + 3 ( 0.99237 ) + 3 cos ( 0.4 ( 0.65784 + 0.6 ( 0.99237 ) ) ) ] = 0.61784 + 0.1 [ 0.6 + 2.97711 + 3 cos ( 0.501304 ) ] And, y 2 = 0.61784 + 0.1 [ 0.6 + 2.97711 + 2.63086 ] = 0.61784 + 0.1 ( 6.20797 ) = 0.61784 + 0.620797 = 1.23864 Substitute 2 for n in equation ( 3 ) . y 3 = y 2 + h 2 [ 2 x 2 + h + 3 cos ( x 2 y 2 ) + 3 cos ( ( x 2 + h ) ( y 2 + h x 2 + 3 h cos ( x 2 y 2 ) ) ) ] ... ( 6 ) Substitute 0.2 for h , 0.4 for x 2 and 1.23864 for y 2 in equation ( 6 ) . y 3 = 1.23864 + 0.2 2 [ 2 ( 0.4 ) + 0.2 + 3 cos ( 0.4 ( 1.23864 ) ) + 3 cos ( ( 0.4 + 0.2 ) ( 1.23864 + 0.2 ( 0.4 ) + 3 ( 0.2 ) cos ( 0

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

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ISBN: 9780137394524

Author: R. Nagle, Edward Saff

Publisher: PEARSON+

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Chapter 3.6, Problem 9E

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The approximate solution to y′=x+3cos(xy) using improved Euler’s method and a rough diagram of the solution on [0,2].

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Chapter 3.6, Problem 8E

Chapter 3.6, Problem 10E

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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

Ch. 3.2 - Prob. 11E Ch. 3.2 - For the logistic curve15, assume pa:=p(ta) and...Ch. 3.2 - In Problem 9, suppose we have the additional...Ch. 3.2 - Prob. 14E Ch. 3.2 - Prob. 15E Ch. 3.2 - 16 Show that for a differentiable function p(t),...Ch. 3.2 - Prob. 18E Ch. 3.2 - Prob. 19E Ch. 3.2 - Prob. 20E Ch. 3.2 - A snowball melts in such a way that the rate of...Ch. 3.2 - Prob. 22E Ch. 3.2 - Prob. 23E Ch. 3.2 - Prob. 24E Ch. 3.2 - Prob. 25E Ch. 3.2 - Prob. 26E Ch. 3.2 - Prob. 27E Ch. 3.3 - Prob. 1E Ch. 3.3 - Prob. 2E Ch. 3.3 - Prob. 3E Ch. 3.3 - Prob. 4E Ch. 3.3 - Prob. 5E Ch. 3.3 - Prob. 6E Ch. 3.3 - Prob. 7E Ch. 3.3 - Prob. 8E Ch. 3.3 - Prob. 9E Ch. 3.3 - Early Monday morning, the temperature in the...Ch. 3.3 - During the summer the temperature inside a van...Ch. 3.3 - Prob. 12E Ch. 3.3 - Prob. 13E Ch. 3.3 - Prob. 14E Ch. 3.3 - Prob. 15E Ch. 3.3 - Prob. 16E Ch. 3.4 - Prob. 1E Ch. 3.4 - Prob. 2E Ch. 3.4 - Prob. 3E Ch. 3.4 - Prob. 4E Ch. 3.4 - Unless otherwise stated, in the following problems...Ch. 3.4 - Unless otherwise stated, in the following problems...Ch. 3.4 - Prob. 7E Ch. 3.4 - Unless otherwise stated, in the following problems...Ch. 3.4 - Prob. 9E Ch. 3.4 - Unless otherwise stated, in the following problems...Ch. 3.4 - Prob. 11E Ch. 3.4 - Prob. 12E Ch. 3.4 - Prob. 13E Ch. 3.4 - Prob. 14E Ch. 3.4 - Prob. 15E Ch. 3.4 - Prob. 16E Ch. 3.4 - In Problem 16, let I=50 kg-m2 and the retarding...Ch. 3.4 - Prob. 18E Ch. 3.4 - Prob. 19E Ch. 3.4 - Prob. 20E Ch. 3.4 - Prob. 21E Ch. 3.4 - Prob. 22E Ch. 3.4 - Prob. 23E Ch. 3.4 - Rocket Flight. A model rocket having initial mass...Ch. 3.4 - Escape Velocity. According to Newtons law of...Ch. 3.5 - An RL circuit with a 5- resistor and a 0.05-H...Ch. 3.5 - Prob. 2E Ch. 3.5 - The pathway for a binary electrical signal between...Ch. 3.5 - If the resistance in the RL circuit of Figure...Ch. 3.5 - Prob. 5E Ch. 3.5 - 6. Derive a power balance equation for the RL and...Ch. 3.5 - 7. An industrial electromagnet can be modeled as...Ch. 3.5 - 8. A 108F capacitor 10 nanofarads is charged to 50...Ch. 3.6 - Prob. 1E Ch. 3.6 - Prob. 2E Ch. 3.6 - Prob. 3E Ch. 3.6 - In Example 1, page 126, the improved Eulers method...Ch. 3.6 - Prob. 5E Ch. 3.6 - Prob. 6E Ch. 3.6 - Prob. 7E Ch. 3.6 - Use the improved Eulers method subroutine with...Ch. 3.6 - Prob. 9E Ch. 3.6 - Prob. 10E Ch. 3.6 - Use the improved Eulers method with tolerance to...Ch. 3.6 - Prob. 12E Ch. 3.6 - Prob. 13E Ch. 3.6 - Prob. 14E Ch. 3.6 - The solution to the initial value problem...Ch. 3.6 - Prob. 16E Ch. 3.6 - Prob. 17E Ch. 3.6 - Prob. 18E Ch. 3.6 - Prob. 20E Ch. 3.7 - Determine the recursive formulas for the Taylor...Ch. 3.7 - Determine the recursive formulas for the Taylor...Ch. 3.7 - Prob. 3E Ch. 3.7 - Prob. 4E Ch. 3.7 - Prob. 5E Ch. 3.7 - Prob. 6E Ch. 3.7 - Prob. 7E Ch. 3.7 - Prob. 8E Ch. 3.7 - Prob. 9E Ch. 3.7 - Prob. 10E Ch. 3.7 - Prob. 11E Ch. 3.7 - Prob. 12E Ch. 3.7 - Prob. 13E Ch. 3.7 - Prob. 14E Ch. 3.7 - Prob. 15E Ch. 3.7 - Prob. 16E Ch. 3.7 - The Taylor method of order 2 can be used to...Ch. 3.7 - Prob. 18E Ch. 3.7 - Prob. 19E Ch. 3.7 - Prob. 20E Ch. 3.7 - Prob. 21E

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The approximate solution to y ′ = x + 3 cos ( x y ) using improved Euler’s method and a rough diagram of the solution on [ 0 , 2 ] .

Solution Summary: The author explains the approximate solution to yprime using improved Euler's method and a rough diagram of the solution.

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To find:

The approximate solution to y′=x+3cos(xy) using improved Euler’s method and a rough diagram of the solution on [0,2].

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The approximate solution to y ′ = x + 3 cos ( x y ) using improved Euler’s method and a rough diagram of the solution on [ 0 , 2 ] .

Solution Summary: The author explains the approximate solution to yprime using improved Euler's method and a rough diagram of the solution.

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To find: The approximate solution to y′=x+3cos(xy) using improved Euler’s method and a rough diagram of the solution on [0,2].

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Chapter 3 Solutions

To find:

The approximate solution to y′=x+3cos(xy) using improved Euler’s method and a rough diagram of the solution on [0,2].