Explanation Given: The given differential equation is, y ′ = 1 − y y ( 0 ) = 0 Formula used: The fourth-order Runge-Kutta method is. y = y 0 + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) Where, k 1 = h f ( x , y ) k 2 = h f ( x + h 2 , y + k 1 2 ) k 3 = h f ( x + h 2 , y + k 2 2 ) k 1 = h f ( x + h , y + k 3 ) Calculation: Consider the differential equation. y ′ = 1 − y …… ( 1 ) The differential equation is the function of x and y so, y ′ = f ( x , y ) , y ( x 0 ) = y 0 …… ( 2 ) Compare equation ( 1 ) and equation ( 2 ) . f ( x , y ) = 1 − y x 0 = 0 y o = 0 Consider the equation of k 1 . k 1 = h ( 1 − y ) ……. ( 3 ) Consider the equation of k 2 . k 2 = h f ( x + h 2 , y + k 1 2 ) k 2 = h ( 1 − ( y + k 1 2 ) ) ……. ( 4 ) Consider the equation of k 3 . k 3 = h f ( x + h 2 , y + k 2 2 ) k 3 = h ( 1 − ( y + k 2 2 ) ) ……. ( 5 ) Consider the equation of k 4 . k 4 = h ( 1 − ( 1 + k 4 ) ) ……. ( 6 ) Consider fourth-order Runge-Kutta method. y = y 0 + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) …… ( 7 ) Substitute 0.25 for h and 0 for y in equation ( 3 ) . k 1 = 0.25 ( 1 − 0 ) = 0.25 Substitute 0.25 for h , 0.25 for k 1 and 0 for y in equation ( 4 ) . k 2 = 0.25 ( 1 − ( 0 + 0.25 2 ) ) = 0.21875 Substitute 0.25 for h , 0.21875 for k 2 and 0 for y in equation ( 5 ) . k 3 = 0.25 ( 1 − ( 0 + 0.21875 2 ) ) = 0.22265 Substitute 0.25 for h , 0.22265 for k 3 and 0 for y in equation ( 6 ) . k 4 = 0.25 ( 1 − ( 1 + 0.22265 ) ) = 0.1943359 Substitute 1 for y 0 , 0.25 for k 1 , 0.21875 for k 2 , 0.22265 for k 3 , and 0.1943359 for k 4 in equation ( 7 ) . y ( 1 ) = 0 + 1 6 ( 0.25 + 2 ( 0.21875 ) + 2 ( 0.22265 ) + 0.1943359 ) = 0.221191 Substitute 0.25 for h and 0.221191 for y in equation ( 3 ) . k 1 = 0.25 ( 1 − 0.221191 ) = 0.1947 Substitute 0.25 for h , 0.1947 for k 1 and 0.221191 for y in equation ( 4 ) . k 2 = 0.25 ( 1 − ( 0.221191 + 0.1947 2 ) ) = 0.17036 Substitute 0.25 for h , 0.17036 for k 2 and 0.221191 for y in equation ( 5 )

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 3.7, Problem 8E

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The solution for the initial value problem using fourth-order Runge-Kutta method.

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Chapter 3.7, Problem 7E

Chapter 3.7, Problem 9E

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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 solution for the initial value problem using fourth-order Runge-Kutta method.

Solution Summary: The author explains that the fourth-order Runge-Kutta method solves the initial value problem using y(4)=0.632.

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To find: The solution for the initial value problem using fourth-order Runge-Kutta method.

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Chapter 3 Solutions

To find:

The solution for the initial value problem using fourth-order Runge-Kutta method.