order reduce by substitution

a technique to solve higher order differential equation by setting additional variable to substitute. Then solve it by normal ODE method. Lastly substitute it back

Example

$$(2-t)u^{\prime \prime \prime }+(3-t)u^{\prime \prime }=0$$

where $t<2$
consider $z=u^{\prime \prime }$

$$\begin{array}{c}\begin{array}{rl}(2-t)z^{\prime }+(3-t)z& =0\\ z^{\prime }& =\frac{t-3}{2-t}z\\ z^{\prime }\frac{1}{z}& =\frac{t-3}{2-t}\\ z& =C{e}^{\int \frac{t-3}{2-t}dt}\\ z& =C(t-2)e^{-t}\end{array}\end{array}$$

then

$$\begin{array}{c}\begin{array}{rl}{u}^{\prime \prime }& =C(t-2)e^{-t}\\ u^{\prime }& =-C_1{e}^{-t}(t-1)+C_2\\ u& =-C_1{e}^{-t}t+C_3+C_{2}t\end{array}\end{array}$$

$\square$