1.1.3.14 Example 14 \(y^{\prime }=\frac {1}{x^{2}}\)
\[ y^{\prime }=\frac {1}{x^{2}}\]
This is the same as above problem where we found
\[ y_{h}=a_{0}\]
To find
\(y_{p}\) we will use the balance
equation (*) from the above problem which is
\[ rc_{0}x^{r-1}=x^{-2}\]
Hence
\(r-1=-2\) or
\(r=-1\). Therefore
\(rc_{0}=1\) or
\(c_{0}=-1\). The particular
solution is therefore
\[ y_{p}=-x^{-1}\]
Hence the solution is
\begin{align*} y & =y_{h}+y_{p}\\ & =c_{1}\left ( 1+O\left ( x^{2}\right ) \right ) -\frac {1}{x}\end{align*}