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 = 0.160956 (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 = 4.642 (sec), leaf count = 36
\[\left [y \left (x \right ) = x^{\frac {2 \sqrt {3}}{3}} \textit {\_C1} x, y \left (x \right ) = 0, y \left (x \right ) = \frac {x^{2} \textit {\_C1}^{2}}{\textit {\_C2}}+\textit {\_C1} x +\textit {\_C2}\right ]\] 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))
Maple raw output
[y(x) = x^(2/3*3^(1/2))*_C1*x, y(x) = 0, y(x) = x^2*_C1^2/_C2+_C1*x+_C2]