Look for a solution in the form:
#y=cx^n#
then the equation becomes:
#cnx^(n-1) + cx^n/x = c^6x^(6n+2)#
#(cn+c) x^(n-1) = c^6x^(6n+2)#
#n+1 = c^5x^(5n+3)#
As the left hand side is constant, the right hand side must be constant for any #x#, which is possible only if:
#5n+3 = 0#
#n=-3/5#
then:
#c^5 = -3/5+1 = 2/5#
#c= root(5)(2/5)#
and the solution is:
#y = root(5)(2/5)x^(-3/5)#
In fact:
#d/dx( root(5)(2/5)x^(-3/5)) +root(5)(2/5)x^(-3/5)/x = -3/5root(5)(2/5)x^(-8/5) + root(5)(2/5)x^(-8/5)#
#d/dx( root(5)(2/5)x^(-3/5)) +root(5)(2/5)x^(-3/5)/x = (1-3/5) root(5)(2/5)x^(-8/5)#
#d/dx( root(5)(2/5)x^(-3/5)) +root(5)(2/5)x^(-3/5)/x = 2/5 root(5)(2/5)x^(-8/5)#
#d/dx( root(5)(2/5)x^(-3/5)) +root(5)(2/5)x^(-3/5)/x = x^2(root(5)(2/5))^6x^(-18/5)#
#d/dx( root(5)(2/5)x^(-3/5)) +root(5)(2/5)x^(-3/5)/x = x^2(root(5)(2/5)x^(-3/5))^6#
