Let's assume that "y_2=x^{-1}v(x)" is the second linearlyindependent solution of the equation.
Substituting it to the equation leads us to
"2x^2(v''x^{-1}-2v'x^{-2}+2vx^{-3})+3x(v'x^{-1}-vx^{-2})-vx^{-1}=0"
Since "x^{-1}" is the solution of the equation, we get
"2x^2(v''x^{-1}-2v'x^{-2})+3xv'x^{-1}=0"
"2xv''-4v'+3v'=0"
"v''-\\frac{v'}{2x}=0"
Assume
"v'=w"
We get
"w'-\\frac{w}{2x}=0"
"\\frac{dw}{w}=\\frac{dx}{2x}"
"w=\\sqrt{x}+c_1"
Therefore
"v=\\int(\\sqrt{x}+c_1)dx=\\frac{2}{3}x^{3\/2}+c_1x+c_2"
Assuming "c_1=c_2=0" we get the second linearly independent solution in form "y_2=\\frac{2}{3}x^{2\/3}x^{-1}=\\frac{2}{3}x^{-1\/3}"
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