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Throughout this WeBWork problem, let $a,b,$ and $c$ be real numbers with $a \neq 0$. In this exercise we will discover a method that will allow us to solve linear homogeneous differential equations of the form $$ax^2\frac{d^2y}{dx^2}+bx\frac{dy}{dx}+cy=0.$$ An equation of this type, is called a Cauchy-Euler equation.

To solve this equation, we will assume that the solution $y$ of the differential equation is of the form $y=x^{m}$ where $m$ is some number that we need to find. Since $x^2 = 0$ when $x=0$ a solution to a Cauchy -Euler equation will be valid for $x \in (-\infty,0)$ or for $x\in(0,\infty)$. Throughout this assignment, we will assume that $x\in(0,\infty)$. If we substitute $y=x^{m}$ into $ax^2\frac{d^2y}{dx^2}+bx\frac{dy}{dx}+cy=0$, we find that