Explanation Given: The differential equation and function, x 2 y ″ + ( x 2 − x ) y ′ + ( 1 − x ) y = 0 y 1 ( x ) = x Formula used: If the general form the equation y ″ + P y ′ + Q y and one of the solutions is y 1 ( x ) then integral defined function, y 2 ( x ) = y 1 ( x ) ∫ x 0 x e − ∫ P d x y 1 2 d x Calculation: Consider differential equation and function, x 2 y ″ + ( x 2 − x ) y ′ + ( 1 − x ) y = 0 y 1 ( x ) = x Rewrite the differential equation in the standard form and then divide the differential equation x 2 y ″ + ( x 2 − x ) y ′ + ( 1 − x ) y = 0 by x 2 , x 2 y ″ + ( x 2 − x ) y ′ + ( 1 − x ) y = 0 y ″ + ( x 2 − x ) x 2 y ′ + ( 1 − x ) x 2 y = 0 First of all, compare the differential equation y ″ + ( x 2 − x ) x 2 y ′ + ( 1 − x ) x 2 y = 0 to the standard form y ″ + P y ′ + Q y , y ″ + ( x 2 − x ) x 2 y ′ + ( 1 − x ) x 2 y = 0 y 1 = x After the comparison we get, y 1 = x , P ( x ) = ( x 2 − x ) x 2 = 1 − 1 x Find the second solution, y 2 ( x ) = y 1 ∫ e − ∫ P d x y 1 2 d x =</

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

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 4.2, Problem 21E

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The second solution $y2(x)$ of the given differential equation.

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Chapter 4.2, Problem 20E

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Find the indefinite integral. Check Answer: 7x 4 + 1x dx

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

Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - In Problems 14 the given family of functions is...Ch. 4.1 - Given that y=c1+c2x2 is a two-parameter family of...Ch. 4.1 - Find two members of the family of solutions in...Ch. 4.1 - Prob. 7E Ch. 4.1 - Use the general solution of x + 2x = 0 given in...Ch. 4.1 - In Problems 9 and 10 find an interval centered...Ch. 4.1 - In Problems 9 and 10 find an interval centered...

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