Explanation Given: The second order differential equation is y ″ − ( 2 x + 1 ) y = 1 with the initial conditions as y ( 0 ) = 3 and y ′ ( 0 ) = 1 . The step size for one step is h = 0.2 and the step size for two step is h = 0.1 . Concept used: Euler’s method The approximated solution for the first order linear differential equation of the form y ′ = f ( x , y ) with initial value as y ( x 0 ) = y 0 with a step size of h is given as follows: y n + 1 = y n + h f ( x n , y n ) , here x n = x 0 + n h . Calculation: The second order differential equation is given as follows: y ″ − ( 2 x + 1 ) y = 1 … … ( 1 ) As per the Euler’s method the approximated solution for the first order linear differential equation of the form y ′ = f ( x , y ) with initial value as y ( x 0 ) = y 0 with a step size of h is given as follows: y n + 1 = y n + h f ( x n , y n ) … … ( 2 ) The value of x n with each successive step is calculated as follows: x n = x 0 + n h … … ( 3 ) The initial values are given as follows y ( 0 ) = 3 and y ′ ( 0 ) = 1 . This implies that for x 0 = 0 , y 0 = 3 and y ′ 0 = 1 . Obtain the approximated value of y ( 0.2 ) for step size h = 0.2 . For n = 0 the value of x 0 is 0 and the value of y 0 is 3 . To obtain the value of x 1 substitute 0 for x 0 , 1 for n and 0.2 for h in equation ( 3 ) . x 1 = 0 + 1 ⋅ ( 0.2 ) = 0.2 To calculate the value of y 1 or the value of y ( 0.2 ) substitute 0 for n and 0.2 for h in equation ( 2 ) . y 1 = y 0 + ( 0.2 ) f ( x 0 , y 0 ) … … ( 4 ) As per the concept of Euler’s method y ′ = f ( x , y ) so the value of f ( x 0 , y 0 ) is equal to the value of y ′ ( 0 ) and the value of y ′ ( 0 ) is 1 . Substitute 3 for y 0 , and 1 for f ( x 0 , y 0 ) in equation ( 4 ) . y 1 = 3 + ( 0.2 ) ⋅ ( 1 ) = 3 + 0.2 = 3.2 Therefore, the approximated value of y ( 0.2 ) is 3.2 . The value of step size for two step is 0.1 . Obtain the approximated value of y ( 0.2 ) for step size h = 0.1 . To obtain the value of x 1 substitute 0 for x 0 , 1 for n and 0

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

11th Edition

ISBN: 9781305965720

Author: Dennis G. Zill

Publisher: Cengage Learning

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The approximated value of y(0.2), for the equation y″−(2x+1)y=1, y(0)=3 and y′(0)=1 with h=0.2 for one step and h=0.1 for two steps.

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A First Course in Differential Equations with Modeling Applications (MindTap Course List)

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The approximated value of y ( 0.2 ) , for the equation y ″ − ( 2 x + 1 ) y = 1 , y ( 0 ) = 3 and y ′ ( 0 ) = 1 with h = 0.2 for one step and h = 0.1 for two steps.

Solution Summary: The author explains the approximated value of y(0.2) for the equation and the step size for two steps.

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The approximated value of y(0.2), for the equation y″−(2x+1)y=1, y(0)=3 and y′(0)=1 with h=0.2 for one step and h=0.1 for two steps.

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