ODE
\[ x^3 y(x) y'''(x)+3 x^3 y'(x) y''(x)+9 x^2 y(x) y''(x)+9 x^2 y'(x)^2+18 x y(x) y'(x)+3 y(x)^2=0 \] ODE Classification
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.495602 (sec), leaf count = 60
\[\left \{\left \{y(x)\to -\frac {\sqrt {c_1 x^2+c_3 x+2 c_2}}{x^{3/2}}\right \},\left \{y(x)\to \frac {\sqrt {c_1 x^2+c_3 x+2 c_2}}{x^{3/2}}\right \}\right \}\]
Maple ✓
cpu = 0.482 (sec), leaf count = 52
\[\left [y \left (x \right ) = \frac {\sqrt {-x \left (x^{2} \textit {\_C1} +2 \textit {\_C3} x -2 \textit {\_C2} \right )}}{x^{2}}, y \left (x \right ) = -\frac {\sqrt {-x \left (x^{2} \textit {\_C1} +2 \textit {\_C3} x -2 \textit {\_C2} \right )}}{x^{2}}\right ]\] Mathematica raw input
DSolve[3*y[x]^2 + 18*x*y[x]*y'[x] + 9*x^2*y'[x]^2 + 9*x^2*y[x]*y''[x] + 3*x^3*y'[x]*y''[x] + x^3*y[x]*y'''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[x^2*C[1] + 2*C[2] + x*C[3]]/x^(3/2))}, {y[x] -> Sqrt[x^2*C[1] + 2*C[2] + x*C[3]]/x^(3/2)}}
Maple raw input
dsolve(x^3*y(x)*diff(diff(diff(y(x),x),x),x)+3*x^3*diff(y(x),x)*diff(diff(y(x),x),x)+9*x^2*y(x)*diff(diff(y(x),x),x)+9*x^2*diff(y(x),x)^2+18*x*y(x)*diff(y(x),x)+3*y(x)^2 = 0, y(x))
Maple raw output
[y(x) = 1/x^2*(-x*(_C1*x^2+2*_C3*x-2*_C2))^(1/2), y(x) = -1/x^2*(-x*(_C1*x^2+2*_C3*x-2*_C2))^(1/2)]