Find the homogeneous solution to the Cauchy-Euler second order differential equation

Find the homogeneous solution to the Cauchy-Euler second order differential equation

Aboard the M-theory laboratory spaceship, Augustin-Louis Cauchy and Leonhard Euler, equipped with nothing but pen, paper, and their brilliant minds, tackle the second order differential equation through a methodical approach. They start by recognizing the equation as a Cauchy-Euler form, indicative of its characteristic roots solving method. Assuming a solution of the form y=x^m, they substitute this back into the equation to derive a characteristic polynomial. The key step involves calculating the roots of this polynomial, leading them to a general solution in terms of these roots. Through their collaboration, they artfully navigate the complexities of the equation, employing a mix of intuition and mathematical precision, ultimately deriving a solution that showcases the beauty of mathematical problem-solving and the power of intellectual collaboration in the serene, high-tech confines of their futuristic workspace.