Explanation Given Information: Set of functions is given by 2 x 2 y ″ + 5 x y ′ + y = x 2 − x , y = c 1 x − 1 2 + c 2 x − 1 + 1 15 x 2 − 1 6 x . Formula used: Wronskian: Suppose each of the functions f 1 ( x ) , f 2 ( x ) , ⋯ , f n ( x ) possesses at least n − 1 derivative and the determinant. | f 1 f 2 ⋯ , f n f ′ 1 f ′ 2 ⋯ , f ′ n f 1 ′ ′ f 1 ′ ′ ⋯ , f ″ n f 1 n − 1 f 2 n − 1 ⋯ , f n n − 1 | Where the prime denotes the derivative, is called the Wronskian function. Fundamental set of solution: Any set of linearly independent solutions of the homogenous linear equation n -order differential equation a n ( x ) d n y d x n + a n − 1 ( x ) d n − 1 y d x n − 1 + ⋯ + a 1 ( x ) d y d x = 0 on an interval I is said to fundamental set of solution in the interval. Calculation: According to the definition of wronskian, Suppose each of the functions f 1 ( x ) , f 2 ( x ) , … , f n ( x ) possesses at least n − 1 derivative and the determinant. | f 1 f 2 ⋯ , f n f ′ 1 f ′ 2 ⋯ , f ′ n f 1 ′ ′ f 1 ′ ′ ⋯ , f ″ n f 1 n − 1 f 2 n − 1 ⋯ , f n n − 1 | Where the prime denotes the derivative, is called the Wronskian function. First of all, differentiate the equation y 1 = x − 1 2 , y 2 = x − 1 with the respect of x , to apply the Wronskian, y 1 = x − 1 2 , y ′ 1 = − 1 2 x − 3 2 , y ″ 1 = 3 4 x − 5 2 y 2 = x − 1 , y ′ 2 = − x − 2 , y ″ 2 = 2 x − 3 Substitute the value y 1 = x − 1 2 , y ′ 1 = − 1 2 x − 3 2 , y ″ 1 = 3 4 x − 5 2 in the equation 2 x 2 y ″ 1 + 5 x y ′ 1 + y 1 = 0 , 2 x 2 y ″ 1 + 5 x y ′ 1 + y 1 = 0 2 x 2 3 4 x − 5 2 + 5 x − 1 2 x − 3 2 + x − 1 2 = 0 3 2 x − 1 2 − 5 2 x − 1 2 + x − 1 2 = 0 3 x − 1 2 − 5 x − 1 2 + 2 x − 1 2 2 = 0 0 2 = 0 0 = 0 [ True ] Substitute the value y 2 = x − 1 , y

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 3.1, Problem 34E

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To verify: Whether the two parameter family of functions 2x2y″+5xy′+y=x2−x is the general solution of the non-homogenous differential equation y=c1x−12+c2x−1+115x2−16x on the indicated interval (0,∞).

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Chapter 3.1, Problem 33E

Chapter 3.1, Problem 35E

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To verify: Whether the two parameter family of functions 2 x 2 y ″ + 5 x y ′ + y = x 2 − x is the general solution of the non-homogenous differential equation y = c 1 x − 1 2 + c 2 x − 1 + 1 15 x 2 − 1 6 x on the indicated interval ( 0 , ∞ ) .

Solution Summary: The author explains Wronskian's formula for verifying if the two parameter family of functions 2x2y′′+5x

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To verify: Whether the two parameter family of functions 2x2y″+5xy′+y=x2−x is the general solution of the non-homogenous differential equation y=c1x−12+c2x−1+115x2−16x on the indicated interval (0,∞).

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