Explanation Given: The differential equation is in the form y ″ + P ( x ) y ′ + Q ( x ) = f ( x ) and y 1 , y 2 are the linearly independent solutions of the above differential equation. Calculation: The given second order non homogenous differential equation is as follows: y ″ + P ( x ) y ′ + Q ( x ) = f ( x ) It is given that y 1 and y 2 are the linearly independent solution of the homogenous part of the above equation. Substitute v for y ′ in the above equation. v ′ + P ( x ) v + Q ( x ) = f ( x ) v ′ = − P ( x ) v − Q ( x ) + f ( x ) The system of differential formed is as follows: y ′ = v v ′ = − P ( x ) v − Q ( x ) + f ( x ) The above system of equation is represented in the form of matrix as follows: ( y v ) ′ = ( 0 1 − Q − P ) ( y v ) + ( 0 f ) The particular solution of the above equation is calculated as follows: X P = Φ ( x ) ∫ Φ − 1 ( x ) F ( x ) d x Where, Φ ( x ) = ( y 1 y 2 y 1 ′ y 2 ′ ) and F ( x ) = ( 0 f ) . The determinant of the matrix Φ is calculated as follows: | Φ | = | y 1 y 2 y 1 ′ y 2 ′ | = y 1 y 2 ′ − y 1 ′ y 2 The inverse of the matrix Φ is as follows: Φ − 1 ( x ) = 1 y 1 y 2 ′ − y 1 ′ y 2 ( y 2 ′ − y 2 − y 1 ′ y 1 ) Therefore, the particular integral X P is obtained as follows: X P = ( y 1 y 2 y 1 ′ y 2 ′ ) ∫ 1 y 1 y 2 ′ − y 1 ′ y 2 ( y 2 ′ − y 2 − y 1 ′ y 1 ) ( 0 f ) d x Replace the term y 1 y 2 ′ − y 1 ′ y 2 by W in the above equation

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Author: ZILL

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Chapter 8.3, Problem 36E

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To show: In case of non-homogenous differential equation the particular solution of the form XP=Φ(x)∫Φ−1(x)F(x)dx reduces to a form of variation of parameters.

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Chapter 8.3, Problem 35E

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In case of non-homogenous differential equation the particular solution of the form X P = Φ ( x ) ∫ Φ − 1 ( x ) F ( x ) d x reduces to a form of variation of parameters.

Solution Summary: The author explains that non-homogenous differential equations reduce to a form of variation of parameters.

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To show: In case of non-homogenous differential equation the particular solution of the form XP=Φ(x)∫Φ−1(x)F(x)dx reduces to a form of variation of parameters.

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