4.3.2.1 Collection of special transformations

These are special transformation that do not fit in any other type.

1. For ode of form \(\left ( 1-x^{2}\right ) y^{\prime \prime }-xy^{\prime }+y=0\) use \(x=\sin z\). This tranforms the ode to \(y^{\prime \prime }\left ( z\right ) +y\left ( z\right ) =0\).
2. For ode of form \(y^{\prime \prime }+\frac {2x}{1+x^{2}}y^{\prime }+\frac {1}{\left ( 1+x^{2}\right ) ^{2}}y=0\) use transformation \(x=\tan z\) this transforms the ode to \(y^{\prime \prime }\left ( z\right ) +y\left ( z\right ) =0\) as well.
3. For ode of form \(\left ( 1+y^{2}\right ) y^{\prime \prime }-\left ( 2y-1\right ) \left ( y^{\prime }\right ) ^{2}+3x\left ( 1+y^{2}\right ) y^{\prime }=0\) use transformation \(y\left ( x\right ) =\tan \left ( z\left ( x\right ) \right ) \) which gives \(z^{\prime \prime }\left ( x\right ) +\left ( z^{\prime }\left ( x\right ) \right ) ^{2}+3xz^{\prime }\left ( x\right ) =0\).
4. For ode of form \(y^{\prime \prime }\left ( x\right ) -\frac {x}{1-x^{2}}y^{\prime }+\frac {y}{1-x^{2}}=0\) use \(x=\cos \left ( z\right ) \) which gives \(y^{\prime \prime }\left ( z\right ) +y\left ( z\right ) =0\)

Reference: Short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907.