Method of reduction of order, Mathematics

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Consider the equation

x2y′′+ xy′- y = 4x ln x

(a) Verify that x is a solution to the homogeneous equation.

(b) Use the method of reduction of order to derive the second solution to the homogeneous equation as Cx-1

(c) Use the method of variation of parameters to show that the general solution of the equation can be written as

y = x(ln x)2- x ln x + c1x + c2x-1


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