What is the solution for y^{''}-(6/x)y^'-(9/(x^2))y=0 ?

The solution for y^{''}-(6/x)y^'-(9/(x^2))y=0 is y=c_{1}x^{(7+sqrt(85))/2}+c_{2}x^{(7-sqrt(85))/2}

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y^{''}-(6/x)y^'-(9/(x^2))y=0

y^{^{''}} - (\frac{6}{x}) y^{^{'}} - (\frac{9}{x ^{2}}) y = 0

y = c_{1} x^{\frac{7 + 85}{2}} + c_{2} x^{\frac{7 - 85}{2}}

Solution steps

y^{''} (x) - (\frac{6}{x}) y^{'} (x) - (\frac{9}{x ^{2}}) y = 0

Solve linear ODE:

y = c_{1} x^{\frac{7 + 85}{2}} + c_{2} x^{\frac{7 - 85}{2}}

y = c_{1} x^{\frac{7 + 85}{2}} + c_{2} x^{\frac{7 - 85}{2}}

x \to - 2 + lim (\frac{- 2}{x ^{2} - 4})

x \to + 0 lim (\frac{1}{x})

x^{^{''}} + k x^{^{'}} + x = 0

\frac{d}{d x} (\frac{3 x ^{2}}{4})

x \to 0 + lim (cot (4 x))

What is the solution for y^{''}-(6/x)y^'-(9/(x^2))y=0 ?
The solution for y^{''}-(6/x)y^'-(9/(x^2))y=0 is y=c_{1}x^{(7+sqrt(85))/2}+c_{2}x^{(7-sqrt(85))/2}