4.13.35 \(a+(x-6 y(x))^2 y'(x)-6 y(x)^2+2 x y(x)=0\)

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 = 0.0139813 (sec), leaf count = 115

\[\left \{\left \{y(x)\to \frac {1}{6} \left (\sqrt [3]{-18 a x+18 c_1-x^3}+x\right )\right \},\left \{y(x)\to \frac {x}{6}+\frac {1}{12} i \left (\sqrt {3}+i\right ) \sqrt [3]{-18 a x+18 c_1-x^3}\right \},\left \{y(x)\to \frac {x}{6}-\frac {1}{12} \left (1+i \sqrt {3}\right ) \sqrt [3]{-18 a x+18 c_1-x^3}\right \}\right \}\]

Maple
cpu = 0.017 (sec), leaf count = 27

\[ \left \{ {x}^{2}y \relax (x ) -6\,x \left (y \relax (x ) \right ) ^{2}+12\, \left (y \relax (x ) \right ) ^{3}+ax+{\it \_C1}=0 \right \} \] 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 + S
qrt[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),'implicit')

Maple raw output

x^2*y(x)-6*x*y(x)^2+12*y(x)^3+a*x+_C1 = 0