4.2069   3x2y′′(x)2 + 4y′(x)2 − 2 (3xy′(x)+ y(x))y′′(x) = 0

ODE

3x2y′′(x)2 + 4y′(x)2 − 2 (3xy′(x)+ y(x))y′′(x) = 0

ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 0.00891545 (sec), leaf count = 24

{{                    } }
         c21x2
   y(x) →  c2  + c1x + c2

Maple
cpu = 0.241 (sec), leaf count = 41

{ √-   (     )                                         }
  -3-   y-(x)                        -C12x2
   2 ln    x   − ln(x)− -C1 = 0,y(x) =  C2   + C1x + -C2

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