Question

Please help me solve the given differential equation using the Cauchy-Euler equation:

xy"+y'=0

Answer 1

Assume \(y=x^r\) is a solution.
\[y=x^r\\
y'=rx^{r-1}\\
y''=r(r-1)x^{r-2}\]
Plug into the equation,
\[x\left(r(r-1)x^{r-2}\right)+rx^{r-1}=0\]
and solve for \(r\).