ODE
\[ 3 x^2 y''(x)^2+4 y'(x)^2-2 \left (3 x y'(x)+y(x)\right ) y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.00891545 (sec), leaf count = 24
\[\left \{\left \{y(x)\to \frac {c_1^2 x^2}{c_2}+c_1 x+c_2\right \}\right \}\]
Maple ✓
cpu = 0.241 (sec), leaf count = 41
\[ \left \{ {\frac {\sqrt {3}}{2}\ln \left ( {\frac {y \left ( x \right ) }{x}} \right ) }-\ln \left ( x \right ) -{\it \_C1}=0,y \left ( x \right ) ={\frac {{{\it \_C1}}^{2}{x}^{2}}{{\it \_C2}}}+{\it \_C1}\,x+{\it \_C2} \right \} \] Mathematica raw input
DSolve[4*y'[x]^2 - 2*(y[x] + 3*x*y'[x])*y''[x] + 3*x^2*y''[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> x*C[1] + (x^2*C[1]^2)/C[2] + C[2]}}
Maple raw input
dsolve(3*x^2*diff(diff(y(x),x),x)^2-2*(3*x*diff(y(x),x)+y(x))*diff(diff(y(x),x),x)+4*diff(y(x),x)^2 = 0, y(x),'implicit')
Maple raw output
1/2*3^(1/2)*ln(y(x)/x)-ln(x)-_C1 = 0, y(x) = _C1^2/_C2*x^2+_C1*x+_C2