ODE
\[ -x^3 y(x)^6+x \left (x^2+y(x)^4+x y(x)^2\right ) y(x)^2 y'(x)-\left (x^2+y(x)^4+x y(x)^2\right ) y'(x)^2+y'(x)^3=0 \] ODE Classification
[_quadrature]
Book solution method
No Missing Variables ODE, Solve for \(y'\)
Mathematica ✓
cpu = 0.0105282 (sec), leaf count = 105
\[\left \{\left \{y(x)\to -\frac {\sqrt [3]{-\frac {1}{3}}}{\sqrt [3]{-c_1-x}}\right \},\left \{y(x)\to \frac {1}{\sqrt [3]{3} \sqrt [3]{-c_1-x}}\right \},\left \{y(x)\to \frac {(-1)^{2/3}}{\sqrt [3]{3} \sqrt [3]{-c_1-x}}\right \},\left \{y(x)\to c_1+\frac {x^3}{3}\right \},\left \{y(x)\to -\frac {2}{2 c_1+x^2}\right \}\right \}\]
Maple ✓
cpu = 0.01 (sec), leaf count = 39
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{-3}+3\,x-{\it \_C1}=0, \left ( y \left ( x \right ) \right ) ^{-1}+{\frac {{x}^{2}}{2}}-{\it \_C1}=0,y \left ( x \right ) ={\frac {{x}^{3}}{3}}+{\it \_C1} \right \} \] Mathematica raw input
DSolve[-(x^3*y[x]^6) + x*y[x]^2*(x^2 + x*y[x]^2 + y[x]^4)*y'[x] - (x^2 + x*y[x]^2 + y[x]^4)*y'[x]^2 + y'[x]^3 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((-1/3)^(1/3)/(-x - C[1])^(1/3))}, {y[x] -> 1/(3^(1/3)*(-x - C[1])^(1/3))}, {y[x] -> (-1)^(2/3)/(3^(1/3)*(-x - C[1])^(1/3))}, {y[x] -> x^3/3 + C[1]}, {y[x] -> -2/(x^2 + 2*C[1])}}
Maple raw input
dsolve(diff(y(x),x)^3-(x^2+x*y(x)^2+y(x)^4)*diff(y(x),x)^2+x*y(x)^2*(x^2+x*y(x)^2+y(x)^4)*diff(y(x),x)-x^3*y(x)^6 = 0, y(x),'implicit')
Maple raw output
y(x) = 1/3*x^3+_C1, 1/y(x)^3+3*x-_C1 = 0, 1/y(x)+1/2*x^2-_C1 = 0