Explanation Formula used: The equation is exact iff P ″ − Q ′ + R = 0 . Calculation: The given equation is x 2 y ″ + x y ′ − y = 0 . Compare the given equation with P ( x ) y ″ + Q ( x ) y ′ + R ( x ) y = 0 and obtain the values of P ( x ) , ( x ) , R ( x ) as shown below: P ( x ) = x 2 , Q ( x ) = x , R ( x ) = − 1 Substitute P ( x ) = x 2 , Q ( x ) = x , R ( x ) = − 1 in P ″ − Q ′ + R = 0 and check for exactness as follows: P ″ − Q ′ + R = ? 0 ( x 2 ) ′ ′ + ( x ) ′ − 1 = ? 0 2 − 1 − 1 = ? 0 0 = 0 Thus, the given equation is exact _ . Since, the differential equation is exact, there exist a function f ( x ) such that [ P ( x ) y ′ ] ′ + [ f ( x ) y ] ′ = 0 . Hence, substitute P ( x ) = x 2 in [ P ( x ) y ′ ] ′ + [ f ( x ) y ] ′ = 0 as, [ P ( x ) y ′ ] ′ + [ f ( x ) y ] ′ = [ x 2 y ′ ] ′ + [ f ( x ) y ] ′ = x 2 y ″ + 2 x y ′ + f ′ ( x ) y + f ( x ) y = x 2 y ″ + ( 2 x + f ( x ) y ′ ) + f ′ ( x

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 3.2, Problem 45P

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Whether the equation x2y″+xy′−y=0,x>0 is exact and to solve the given equation if exactness is satisfied.

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Chapter 3.2, Problem 44P

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Whether the equation x 2 y ″ + x y ′ − y = 0 , x &gt; 0 is exact and to solve the given equation if exactness is satisfied.

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Whether the equation x2y″+xy′−y=0,x>0 is exact and to solve the given equation if exactness is satisfied.

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Whether the equation x 2 y ″ + x y ′ − y = 0 , x > 0 is exact and to solve the given equation if exactness is satisfied.