A third$-$order Euler equation $is$ one $of$ the form

$ax$ cubed $y$ triple prime plus $bx$ squared $y$ double prime plus $cxy$ prime plus kyax$3y+bx2y+cxy+$kyequals$=0,$

where $a,b,c,$ and $k$ are constants. $If$

xgreater than$>0,$

then the substitution

vequals$=lnxlnx$

transforms the equation into the following constant$-$coefficient equation with independent variable $v.$

a StartFraction $d$ cubed $y$ Over $dv$ cubed EndFraction plus left parenthesis $b$ minus $3$ a right parenthesis StartFraction $d$ squared $y$ Over $dv$ squared EndFraction plus left parenthesis $c$ minus $b$ plus $2$ a right parenthesis StartFraction $dy$ Over $dv$ EndFraction plus kyad$3ydv3+(b-3a)d2ydv2+(c-b+2a)dydv+$kyequals$=0$

Make the substitution

vequals$=lnxlnx$

$to$ find the general solution $of$

$x$ squared $y$ double prime plus $xy$ prime plus $9yx2y+xy+9$yequals$=0$

for

xgreater than$>0.$