1.2.2.2.5 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.