Example 2 \[ y^{\prime }+2xy=\frac {1}{x}\] Expansion is around \(x=0\). The (homogenous) ode has the form \(y^{\prime }+p\left ( x\right ) y=0\). We see that \(p\left ( x\right ) \) is defined as is at \(x=0\). However the RHS has no series expansion at \(x=0\). Therefore we must use Frobenius series.  This is the same ode as example 1. So we go straight to find \(y_{p}\) as \(y_{h}\) is the same.  Now we need to find \(y_{p}\) using the balance equation. From above we found that\[ ra_{0}x^{r-1}=\frac {1}{x}\] Renaming \(a\) to \(c\) and \(r\) as \(m\) so not to confuse terms used for \(y_{h}\), the above becomes\[ mc_{0}x^{m-1}=x^{-1}\] Hence \(m-1=-1\) or \(m=0\). Therefore \(mc_{0}=1\). But since \(m=0\) then no solution for \(c_{0}\). Hence it is not possible to find series solution. This is an example where the balance equation fails and so we have to use asymptotic expansion to find solution, which is not supported now.