Explanation Procedure used: “Find an approximate solution of a boundary-value problem.” (1) Compare given differential equation with y ″ + P ( x ) y ′ + Q ( x ) y = f ( x ) to identify the functions P ( x ) , Q ( x ) and f ( x ) ; and the boundary conditions with y ( a ) = α , y ( b ) = β to identify the boundary [ a , b ] . Also, identify the number of steps n . (2) Compute step size of the approximation using the formula h = b − a n and use the value to identify the interior mesh points x i = a + ( i + 1 ) h for i = 0 , 1 , 2 , … , n . (3) At each interior mesh point x i with i = 1 , 2 , … , n − 1 as obtained in Step 2, define p i = P ( x i ) , q i = Q ( x i ) and f i = f ( x i ) . (4) Formulate the system of difference equations for the interior mesh points corresponding to i = 1 , 2 , … , n − 1 as ( 1 + h 2 p i ) y i + 1 + ( − 2 + h 2 q i ) y i + ( 1 − h 2 p i ) y i − 1 = h 2 f i using the values obtained in Step 3. (5) Solve the system using given boundary value conditions to obtain the functional values at each interior mesh point. Calculation: Given differential equation is y ″ + 9 y = 0 with boundary values as y ( 0 ) = 4 and y ( 2 ) = 1 . Write the equation in standard form as, y ″ + 0 y ′ + 9 y = 0 Compare the equation with general form to identify the required functions. P ( x ) = 0 Q ( x ) = 9 f ( x ) = 0 The boundary which is to be partitioned into n = 4 interior mesh points and in which the solution is to be approximated is [ 0 , 2 ] . Compute the value of h . h = 2 − 0 4 = 0.5 Find the interior mesh points and the functional values p i , q i and f i corresponding to each point x i . Calculate the value of x 1 as shown below: x 1 = 0 + ( 1 ⋅ 0.5 ) = 0.5 Calculate the value of p 1 as shown below: p 1 = P ( 0.5 ) = 0 Calculate the value of q 1 as shown below: q 1 = Q ( 0.5 ) = 9 Calculate the value of f 1 as shown below: f 1 = f ( 0.5 ) = 0 This gives that when i = 1 , then p 1 = 0 , q 1 = 9 , f 1 = 0 . The difference equation is expressed as follows: ( 1 + ( 0.5 2 ⋅ 0 ) ) y 1 + 1 + { − 2 + ( ( 0.5 ) 2 ⋅ 9 ) } y 1 + ( 1 − ( 0.5 2 ⋅ 0 ) ) y 1 − 1 = ( 0.5 ) 2 ⋅ 0 y 2 + 0.25 y 1 + y 0 = 0 Calculate the value of x 2 as shown below: x 2 = 0 + ( 2 ⋅ 0.5 ) = 1.0 Calculate the value of p 2 as shown below: p 2 = P ( 1.0 ) = 0 Calculate the value of q 2 as shown below: q 2 = Q ( 1.0 ) = 9 Calculate the value of f 2 as shown below: f 2 = f ( 1.0 ) = 0 This gives that when i = 2 , then p 2 = 0 , q 2 = 9 , f 2 = 0 . The difference equation is expressed as follows: ( 1 + ( 0.5 2 ⋅ 0 ) ) y 2 + 1 + { − 2 + ( ( 0.5 ) 2 ⋅ 9 ) } y 2 + ( 1 − ( 0.5 2 ⋅ 0 ) ) y 2 − 1 = ( 0.5 ) 2 ⋅ 0 y 3 + 0.25 y 2 + y 1 = 0 Calculate the value of x 3 as shown below: x 3 = 0 + ( 3 ⋅ 0

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Differential Equations with Boundary-Value Problems (MindTap Course List)

Ch. 9.1 - In Problems 110 use the improved Eulers method to...Ch. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E Ch. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - In Problems 1–10 use the improved Euler’s method...Ch. 9.1 - Prob. 9E Ch. 9.1 - Prob. 10E

Ch. 9.1 - Prob. 11E Ch. 9.1 - Prob. 13E Ch. 9.1 - Prob. 14E Ch. 9.1 - Prob. 15E Ch. 9.1 - Prob. 16E Ch. 9.1 - Consider the initial-value problem y = 2x 3y + 1,...Ch. 9.1 - Prob. 18E Ch. 9.1 - Prob. 19E Ch. 9.1 - Repeat Problem 19 using the improved Euler’s...Ch. 9.1 - Prob. 21E Ch. 9.2 - Use the RK4 method with h = 0.1 to approximate...Ch. 9.2 - Prob. 2E Ch. 9.2 - Prob. 3E Ch. 9.2 - Prob. 4E Ch. 9.2 - Prob. 5E Ch. 9.2 - Prob. 6E Ch. 9.2 - Prob. 7E Ch. 9.2 - Prob. 8E Ch. 9.2 - Prob. 9E Ch. 9.2 - Prob. 10E Ch. 9.2 - In Problems 312 use the RK4 method with h = 0.1 to...Ch. 9.2 - Prob. 12E Ch. 9.2 - Prob. 13E Ch. 9.2 - Consider the initial-value problem y′ = 2y, y(0) =...Ch. 9.2 - Prob. 17E Ch. 9.2 - Consider the initial-value problem y′ = 2x – 3y +...Ch. 9.2 - Prob. 19E Ch. 9.2 - Prob. 20E Ch. 9.3 - Prob. 1E Ch. 9.3 - Prob. 3E Ch. 9.3 - Prob. 4E Ch. 9.3 - Prob. 5E Ch. 9.3 - Prob. 6E Ch. 9.3 - Prob. 7E Ch. 9.3 - Prob. 8E Ch. 9.4 - Use Euler’s method to approximate y(0.2), where...Ch. 9.4 - Prob. 2E Ch. 9.4 - Prob. 3E Ch. 9.4 - Prob. 4E Ch. 9.4 - Prob. 5E Ch. 9.5 - Prob. 1E Ch. 9.5 - Prob. 2E Ch. 9.5 - Prob. 3E Ch. 9.5 - Prob. 4E Ch. 9.5 - Prob. 5E Ch. 9.5 - In Problems 1-18 use Definition 7.1.1 to find ℒ{f...Ch. 9.5 - Prob. 7E Ch. 9.5 - Prob. 8E Ch. 9.5 - Prob. 9E Ch. 9.5 - Prob. 10E Ch. 9.5 - Prob. 11E Ch. 9.5 - The electrostatic potential u between two...Ch. 9.5 - Consider the boundary-value problem y″ + xy = 0,...Ch. 9 - Prob. 1RE Ch. 9 - Prob. 2RE Ch. 9 - In Problems 1–4 construct a table comparing the...Ch. 9 - In Problems 1–4 construct a table comparing the...Ch. 9 - Prob. 5RE Ch. 9 - Prob. 6RE Ch. 9 - Prob. 7RE Ch. 9 - Prob. 8RE

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