2.2.2.0 Example 5 \(x^{2}y^{\prime \prime }+xy^{\prime }+xy=1\)

\[ x^{2}y^{\prime \prime }+xy^{\prime }+xy=1 \]

The homogenous ode was solved up, so we just need to find \(y_{p}\). To find \(y_{p}\), and using \(c\) in place of \(a\) so not to confuse terms with the \(y_{h}\) terms, then from the above problem, we found the indicial equation. Hence the balance equation is

\[ c_{0}r^{2}x^{r}=1 \]

To balance this we need \(r=0\).  Hence \(0c_{0}=1\) which is not possible. Hence no particular solution exists. No solution in series exists.