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Math Advanced Math Custom Kreyszig: Advanced Engineering Mathematics

The ordinary differential equation ( x 3 D 3 + x 2 D 2 − 2 x D + 2 I ) y = x − 2 . The solution of differential equation is y = ( c 1 x − 1 + c 2 x ) − 1 12 x 2 _ . Formula used: The particular solution of the differential equation is y p ( x ) = y 1 ( x ) ∫ W 1 ( x ) W ( x ) r ( x ) d x + y 2 ( x ) ∫ W 2 ( x ) W ( x ) r ( x ) d x + y 3 ( x ) ∫ W 3 ( x ) W ( x ) r ( x ) d x . Where y 1 , y 2 , y 3 are basis of the homogenous differential equation. Calculation: The characteristic equation of the differential equation is written as: m ( m − 1 ) ( m − 2 ) x m + m ( m − 1 ) x m − 2 m x m + 2 x m = 0 m ( m − 1 ) ( m − 2 ) + m ( m − 1 ) − 2 m + 2 = 0 m 3 − 2 m 2 − m + 2 = 0 On the further simplification, the following obtained. ( m − 1 ) ( m − 2 ) ( m + 1 ) = 0 m = − 1 , 1 , 2 The value of W ( y 1 , y 2 , y 3 ) is calculated as: W ( y 1 , y 2 , y 3 ) = | y 1 y 2 y 3 y 1 ′ y 2 ′ y 3 ′ y 1 ″ y 2 ″ y 3 ″ | = | 1 x x x 2 − 1 x 2 1 2 x 2 x 3 0 2 | = 6 x The value of W 1 is calculated as: W 1 = | 0 x x 2 0 1 2 x 1 0 2 | = x 2 The value of W 2 is calculated as: W 2 = | 1 x 0 x 2 − 1 x 2 0 2 x 2 x 3 1 2 | = − 3 The value of W 3 is calculated as: W 3 = | 1 x x 0 − 1 x 2 1 0 2 x 3 0 1 | = 2 x Now the value of y p is calculated as: y p ( x ) = 1 x ∫ ( x 3 6 ) ( 1 x ) d x + x ∫ ( − 3 x 6 ) ( 1 x ) d x + x 2 ∫ ( 1 3 ) ( 1 x 5 ) d x = 1 6 x ∫ ( 1 x 2 ) d x − x 2 ∫ ( 1 x 4 ) d x + x 2 3 ∫ ( 1 x 5 ) d x = 1 6 x ( − 1 x ) − x 2 ( − 1 3 x 3 ) + x 2 3 ( − 1 4 x 4 ) = ( − 1 12 x 2 ) Therefore, the general solution of the differential equation is written as: y = ( c 1 x − 1 + c 2 x ) + y p = ( c 1 x − 1 + c 2 x ) − 1 12 x 2 Therefore, the general solution of differential equation is y = ( c 1 x − 1 + c 2 x ) − 1 12 x 2 _ .

The ordinary differential equation ( x 3 D 3 + x 2 D 2 − 2 x D + 2 I ) y = x − 2 . The solution of differential equation is y = ( c 1 x − 1 + c 2 x ) − 1 12 x 2 _ . Formula used: The particular solution of the differential equation is y p ( x ) = y 1 ( x ) ∫ W 1 ( x ) W ( x ) r ( x ) d x + y 2 ( x ) ∫ W 2 ( x ) W ( x ) r ( x ) d x + y 3 ( x ) ∫ W 3 ( x ) W ( x ) r ( x ) d x . Where y 1 , y 2 , y 3 are basis of the homogenous differential equation. Calculation: The characteristic equation of the differential equation is written as: m ( m − 1 ) ( m − 2 ) x m + m ( m − 1 ) x m − 2 m x m + 2 x m = 0 m ( m − 1 ) ( m − 2 ) + m ( m − 1 ) − 2 m + 2 = 0 m 3 − 2 m 2 − m + 2 = 0 On the further simplification, the following obtained. ( m − 1 ) ( m − 2 ) ( m + 1 ) = 0 m = − 1 , 1 , 2 The value of W ( y 1 , y 2 , y 3 ) is calculated as: W ( y 1 , y 2 , y 3 ) = | y 1 y 2 y 3 y 1 ′ y 2 ′ y 3 ′ y 1 ″ y 2 ″ y 3 ″ | = | 1 x x x 2 − 1 x 2 1 2 x 2 x 3 0 2 | = 6 x The value of W 1 is calculated as: W 1 = | 0 x x 2 0 1 2 x 1 0 2 | = x 2 The value of W 2 is calculated as: W 2 = | 1 x 0 x 2 − 1 x 2 0 2 x 2 x 3 1 2 | = − 3 The value of W 3 is calculated as: W 3 = | 1 x x 0 − 1 x 2 1 0 2 x 3 0 1 | = 2 x Now the value of y p is calculated as: y p ( x ) = 1 x ∫ ( x 3 6 ) ( 1 x ) d x + x ∫ ( − 3 x 6 ) ( 1 x ) d x + x 2 ∫ ( 1 3 ) ( 1 x 5 ) d x = 1 6 x ∫ ( 1 x 2 ) d x − x 2 ∫ ( 1 x 4 ) d x + x 2 3 ∫ ( 1 x 5 ) d x = 1 6 x ( − 1 x ) − x 2 ( − 1 3 x 3 ) + x 2 3 ( − 1 4 x 4 ) = ( − 1 12 x 2 ) Therefore, the general solution of the differential equation is written as: y = ( c 1 x − 1 + c 2 x ) + y p = ( c 1 x − 1 + c 2 x ) − 1 12 x 2 Therefore, the general solution of differential equation is y = ( c 1 x − 1 + c 2 x ) − 1 12 x 2 _ .

Custom Kreyszig: Advanced Engineering Mathematics

Custom Kreyszig: Advanced Engineering Mathematics

10th Edition

ISBN: 9781119166856

Author: Kreyszig

Publisher: JOHN WILEY+SONS INC.CUSTOM

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Custom Kreyszig: Advanced Engineering Mathematics

Ch. 3.1 - 1–6 BASES: TYPICAL EXAMPLES To get a feel for...Ch. 3.1 - 1–6 BASES: TYPICAL EXAMPLES To get a feel for...Ch. 3.1 - 1–6 BASES: TYPICAL EXAMPLES To get a feel for...Ch. 3.1 - 1–6 BASES: TYPICAL EXAMPLES To get a feel for...Ch. 3.1 - Prob. 5P Ch. 3.1 - Prob. 6P Ch. 3.1 - Prob. 8P Ch. 3.1 - Prob. 9P Ch. 3.1 - Prob. 10P Ch. 3.1 - Prob. 11P

Ch. 3.1 - Prob. 12P Ch. 3.1 - Prob. 13P Ch. 3.1 - Prob. 14P Ch. 3.1 - Prob. 15P Ch. 3.1 - Prob. 16P Ch. 3.2 - Prob. 1P Ch. 3.2 - Prob. 2P Ch. 3.2 - Solve the given ODE. Show the details of your...Ch. 3.2 - Solve the given ODE. Show the details of your...Ch. 3.2 - Solve the given ODE. Show the details of your...Ch. 3.2 - Solve the given ODE. Show the details of your...Ch. 3.2 - Solve the IVP by a CAS, giving a general solution...Ch. 3.2 - Prob. 8P Ch. 3.2 - Solve the IVP by a CAS, giving a general solution...Ch. 3.2 - Solve the IVP by a CAS, giving a general solution...Ch. 3.2 - Solve the IVP by a CAS, giving a general solution...Ch. 3.2 - CAS EXPERIMENT. Reduction of Order. Starting with...Ch. 3.3 - Solve the following ODEs, showing the details of...Ch. 3.3 - Solve the following ODEs, showing the details of...Ch. 3.3 - Solve the following ODEs, showing the details of...Ch. 3.3 - Solve the following ODEs, showing the details of...Ch. 3.3 - Prob. 5P Ch. 3.3 - Prob. 6P Ch. 3.3 - Solve the following ODEs, showing the details of...Ch. 3.3 - Solve the given IVP, showing the details of your...Ch. 3.3 - Solve the given IVP, showing the details of your...Ch. 3.3 - Prob. 10P Ch. 3.3 - Solve the given IVP, showing the details of your...Ch. 3.3 - Solve the given IVP, showing the details of your...Ch. 3.3 - Solve the given IVP, showing the details of your...Ch. 3 - Prob. 1RQ Ch. 3 - List some other basic theorems that extend from...Ch. 3 - If you know a general solution of a homogeneous...Ch. 3 - What form does an initial value problem for an...Ch. 3 - What is the Wronskian? What is it used for? Ch. 3 - Prob. 6RQ Ch. 3 - Solve the given ODE. Show the details of your...Ch. 3 - Solve the given ODE. Show the details of your...Ch. 3 - Prob. 9RQ Ch. 3 - Solve the given ODE. Show the details of your...Ch. 3 - Solve the given ODE. Show the details of your...Ch. 3 - Prob. 12RQ Ch. 3 - Solve the given ODE. Show the details of your...Ch. 3 - Prob. 14RQ Ch. 3 - Prob. 15RQ Ch. 3 - Solve the IVP. Show the details of your work. Ch. 3 - Solve the IVP. Show the details of your work. y‴ +...Ch. 3 - Solve the IVP. Show the details of your work. Ch. 3 - Solve the IVP. Show the details of your work. Ch. 3 - Solve the IVP. Show the details of your work.

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