ODE
\[ a+(x-6 y(x))^2 y'(x)-6 y(x)^2+2 x y(x)=0 \] ODE Classification
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`]]
Book solution method
Exact equation
Mathematica ✓
cpu = 0.459465 (sec), leaf count = 115
\[\left \{\left \{y(x)\to \frac {1}{6} \left (x+\sqrt [3]{-18 a x-x^3+18 c_1}\right )\right \},\left \{y(x)\to \frac {x}{6}+\frac {1}{12} i \left (\sqrt {3}+i\right ) \sqrt [3]{-18 a x-x^3+18 c_1}\right \},\left \{y(x)\to \frac {x}{6}-\frac {1}{12} \left (1+i \sqrt {3}\right ) \sqrt [3]{-18 a x-x^3+18 c_1}\right \}\right \}\]
Maple ✓
cpu = 0.043 (sec), leaf count = 115
\[\left [y \left (x \right ) = \frac {\left (-x^{3}-18 a x -18 \textit {\_C1} \right )^{\frac {1}{3}}}{6}+\frac {x}{6}, y \left (x \right ) = -\frac {\left (-x^{3}-18 a x -18 \textit {\_C1} \right )^{\frac {1}{3}}}{12}-\frac {i \sqrt {3}\, \left (-x^{3}-18 a x -18 \textit {\_C1} \right )^{\frac {1}{3}}}{12}+\frac {x}{6}, y \left (x \right ) = -\frac {\left (-x^{3}-18 a x -18 \textit {\_C1} \right )^{\frac {1}{3}}}{12}+\frac {i \sqrt {3}\, \left (-x^{3}-18 a x -18 \textit {\_C1} \right )^{\frac {1}{3}}}{12}+\frac {x}{6}\right ]\] Mathematica raw input
DSolve[a + 2*x*y[x] - 6*y[x]^2 + (x - 6*y[x])^2*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x + (-18*a*x - x^3 + 18*C[1])^(1/3))/6}, {y[x] -> x/6 + (I/12)*(I + Sqrt[3])*(-18*a*x - x^3 + 18*C[1])^(1/3)}, {y[x] -> x/6 - ((1 + I*Sqrt[3])*(-18*a*x - x^3 + 18*C[1])^(1/3))/12}}
Maple raw input
dsolve((x-6*y(x))^2*diff(y(x),x)+a+2*x*y(x)-6*y(x)^2 = 0, y(x))
Maple raw output
[y(x) = 1/6*(-x^3-18*a*x-18*_C1)^(1/3)+1/6*x, y(x) = -1/12*(-x^3-18*a*x-18*_C1)^(1/3)-1/12*I*3^(1/2)*(-x^3-18*a*x-18*_C1)^(1/3)+1/6*x, y(x) = -1/12*(-x^3-18*a*x-18*_C1)^(1/3)+1/12*I*3^(1/2)*(-x^3-18*a*x-18*_C1)^(1/3)+1/6*x]