Explanation Given: Given differential equation is x 2 y ″ ( x ) + 2 x y ′ ( x ) − 2 y ( x ) = 6 x − 2 + 3 x Approach: Variation of parameters can be used to find particular solution of a second order non homogeneous equation in the form y ″ ( x ) + p ( x ) y ′ ( x ) + q ( x ) y ( x ) = f ( x ) . Formula of variation of parameters to find particular solution is given by, y p ( x ) = − y 1 ( x ) ∫ y 2 ( x ) f ( x ) a ( x ) W ( x ) d x + y 2 ( x ) ∫ y 1 ( x ) f ( x ) a ( x ) W ( x ) d x , where W ( t ) is called Wronskian and defined as W ( x ) = y 1 ( x ) y ′ 2 ( x ) − y ′ 1 ( x ) y 2 ( x ) . Term y 1 and y 2 are linearly independent. A second-order homogeneous equation in the form a x 2 y ″ ( x ) + b x y ′ ( x ) + c y ( x ) = 0 is called Cauchy - Euler equation, where a ≠ 0 and b and c are real numbers. If the equation has a singular point then it has solution in the form of y = x r . If the roots of equation a r 2 + b r + c = 0 are real such that r 1 ≠ r 2 then general solution is y ( t ) = c 1 e r 1 t + c 2 e r 2 t Calculation: Consider the given problem. x 2 y ″ ( x ) + 2 x y ′ ( x ) − 2 y ( x ) = 6 x − 2 + 3 x ... ( 1 ) Let, w ( r , x ) = ( x ) 2 and L is the left hand side of the equation. Differentiate w ( r , x ) = x r with respect to r upto second derivative. d 2 d r 2 { L [ w ] ( x ) } = d 2 d r 2 ( x ) r = d d r r ( x ) x r − 1 = r ( r − 1 ) x r − 2 Substitute r ( r − 1 ) x r − 2 for y ″ , r ( x ) x r − 1 for y ′ , and ( x ) r for y in equation ( 1 ) . L [ w ] ( x ) = [ x 2 r ( r − 1 ) x r − 2 + 2 x r x r − 1 − 2 x r ] = [ r ( r − 1 ) x r + 2 r x r − 2 x r ] = [ r ( r − 1 ) + 2 r − 2 ] ( x ) r = [ r 2 + r − 2 ] ( x ) r Equate the coefficient of ( x ) r with zero. r 2 + r − 2 = 0 ( 2 ) Simplify equation ( 2 ) . r 2 + r − 2 = 0 r 2 + 2 r − r − 2 = 0 r ( r + 2 ) − ( r + 2 ) = 0 ( r + 2 ) ( r − 1 ) = 0 The equation ( 2 ) has two roots − 2 and 1 . As the roots of the equation are distinct and real number therefore the general solution is, y ( x ) = c 1 x − 2 + c 2 x … ( 3 ) From the equation ( 3 ) and equation ( 1 )

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 8.5, Problem 14E

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The general solution of the differential equation for x > 0 by the use of variation of parameters.

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Solution Summary: The author explains how variation of parameters can be used to find the general solution of a second order non homogeneous equation.

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The general solution of the differential equation for x > 0 by the use of variation of parameters.

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The general solution of the differential equation for $x>0$ by the use of variation of parameters.