The point $x=0$ is a regular singular point of the given differential equation.

$36x^2{y}^{\text{'}\text{'}}+36x^2{y}^{\text{'}}+5y=0$

Show that the indicial roots $r$ of the singularity do not differ by an integer. $($List the indicial roots below as a comma$-$separated list.$)$

$r=$

Use the method of Frobenius to obtain two linearly independent series solutions about $x=0.$ Form the general solution on $(0,).$

$y=C_1(1-\frac{x}{6}+\frac{7x^2}{144}-\frac{91x^3}{7776}+$dots$)+C_2{x}^{\frac{1}{6}}(1-\frac{x}{6}+\frac{7x^2}{144}-\frac{91x^3}{7776}+$dots$)$

$y=C_1{x}^{\frac{1}{6}}(1-\frac{x}{2}+\frac{7x^2}{32}-\frac{13x^3}{192}+$dots$)+C_2{x}^{\frac{5}{6}}(1-\frac{x}{2}+\frac{11x^2}{64}-\frac{17x^3}{384}+$dots$)$

$y=C_1{x}^{\frac{1}{6}}(1-\frac{x}{2}+\frac{11x^2}{64}-\frac{17x^3}{384}+$dots$)+C_2{x}^{\frac{5}{6}}(1-\frac{x}{2}+\frac{7x^2}{32}-\frac{13x^3}{192}+$dots$)$

$y=C_1{x}^{\frac{1}{6}}(1+\frac{x}{2}+\frac{7x^2}{32}+\frac{13x^3}{192}+$dots$)+C_2{x}^{\frac{5}{6}}(1+\frac{x}{2}+\frac{11x^2}{64}+\frac{17x^3}{384}+$dots$)$

$y=C_1{x}^{\frac{1}{6}}(1+\frac{x}{2}+\frac{11x^2}{64}+\frac{17x^3}{384}+$dots$)+C_2{x}^{\frac{5}{6}}(1+\frac{x}{2}+\frac{7x^2}{32}+\frac{13x^3}{192}+$dots$)$