Find the general solution of the equation:\(\displaystyle $x^2y'' - 9xy' + 25y = 0, y_1=x^5$\) is a solution.

So I found the second solution with Abel's formula: \(\displaystyle y_2 = y_1 * \int \frac{W}{y_1^2}dx, W = C_0*\exp^\psi, \psi = -\int p(x)\)

Skipping a few steps, I got \(\displaystyle \psi=9*\int \frac{1}{x}dx = \ln(x^9) + 9*C_1\) and \(\displaystyle W = C_0*x^9*\exp^{9*C_1} = C*x^9, C=C_0*\exp^{9*C_1}\)

I eventually found \(\displaystyle y_2\) as \(\displaystyle y_2 = C*x^5*\ln(x) + C*x^5*C_2\)

At this point, I know I need to apply variation of parameters. But I feel like I'm missing something because applying variation of parameters leads to a pretty complicated result.

Is there an easier way to solve this problem? Is it possible to use the characteristic equation to solve it? Any help is appreciated.