This problem has multiple parts. After you submit your answers, you will be given the option to move on to the next part. As you will submit this problem to me, before you move on to the next part, please print your work.

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 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

 $0=ax^2\frac{d^2y}{dx^2}+bx\frac{dy}{dx}+cy$
 $\hskip 48pt =x^{m}\Big[$ $m^2+$ $m +$ $\Big]$

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Note: This problem has more than one part.
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your overall score is for all the parts combined.