Use the equation

$x^2{y}^{\text{'}\text{'}}+x{y}^{\text{'}}+(x^2-v^2)y=0$

to find the general solution of the given differential equation on $(0,).$

$81x^2{y}^{\text{'}\text{'}}+81x{y}^{\text{'}}+(81x^2-1)y=0$

$y(x)=C_1{I}_{\frac{1}{9}}(x)+C_2{K}_{\frac{1}{9}}(x)$

$y(x)=C_1{I}_{\frac{1}{81}}(x)+C_2{K}_{\frac{1}{81}}(x)$

$y(x)=C_1{J}_{\frac{1}{9}}(x)+C_2{J}_{-\frac{1}{9}}(x)$

$y(x)=C_1{J}_1(x)+C_2{J}_{-1}(x)$

$y(x)=C_1{J}_{\frac{1}{81}}(x)+C_2{J}_{-\frac{1}{81}}(x)$