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*}