1.3.1.1.1 Algorithm For the special case of \(n=-2\) the solution can be written directly as given by Eqworld ode0106 as

\begin{equation} y=\frac {\lambda }{x}-\frac {x^{2b\lambda }}{\frac {bx}{2b\lambda +1}x^{2b\lambda }+c_{1}} \tag {1}\end{equation}

Where in the above \(\lambda \) is a root of \(b\lambda ^{2}+\lambda +a=0\).

There is another way to solve the above with \(n=-2\). This can be solved using the substitution

\begin{equation} y=\frac {1}{u} \tag {2}\end{equation}

Hence \(y^{\prime }=-\frac {u^{\prime }}{u^{2}}\) and the ode becomes

\begin{align*} -\frac {u^{\prime }}{u^{2}} & =ax^{-2}+b\frac {1}{u^{2}}\\ -u^{\prime } & =a\frac {u^{2}}{x^{2}}+b\\ u^{\prime } & =-a\frac {u^{2}}{x^{2}}-b \end{align*}

Which is first order Homogeneous ode type (see earlier section). But using (1) is much simpler method as solution can be written directly. The following example shows that using (1) and (2) give same solution.