4.14.36 \(2 y(x)^3 y'(x)=x^3-x y(x)^2\)

ODE
\[ 2 y(x)^3 y'(x)=x^3-x y(x)^2 \] ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.101282 (sec), leaf count = 878

\[\left \{\left \{y(x)\to -\frac {\sqrt {\frac {x^4}{\sqrt [3]{x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}}}-x^2+\sqrt [3]{x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}}}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {x^4}{\sqrt [3]{x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}}}-x^2+\sqrt [3]{x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}}}}{\sqrt {2}}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {\frac {i \left (i+\sqrt {3}\right ) x^4-2 \sqrt [3]{x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}} x^2+\left (-1-i \sqrt {3}\right ) \left (x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}\right ){}^{2/3}}{\sqrt [3]{x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}}}}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {\frac {i \left (i+\sqrt {3}\right ) x^4-2 \sqrt [3]{x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}} x^2+\left (-1-i \sqrt {3}\right ) \left (x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}\right ){}^{2/3}}{\sqrt [3]{x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}}}}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {\frac {\left (-1-i \sqrt {3}\right ) x^4-2 \sqrt [3]{x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}} x^2+i \left (i+\sqrt {3}\right ) \left (x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}\right ){}^{2/3}}{\sqrt [3]{x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}}}}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {\frac {\left (-1-i \sqrt {3}\right ) x^4-2 \sqrt [3]{x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}} x^2+i \left (i+\sqrt {3}\right ) \left (x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}\right ){}^{2/3}}{\sqrt [3]{x^6-2 e^{12 c_1}+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}}}}\right \}\right \}\]

Maple
cpu = 0.023 (sec), leaf count = 45

\[ \left \{ -{\frac {1}{6}\ln \left ({\frac {-{x}^{2}+2\, \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) }-{\frac {1}{3}\ln \left ({\frac {{x}^{2}+ \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[2*y[x]^3*y'[x] == x^3 - x*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> -(Sqrt[-x^2 + x^4/(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C
[1])*x^6])^(1/3) + (-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6]
)^(1/3)]/Sqrt[2])}, {y[x] -> Sqrt[-x^2 + x^4/(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(2
4*C[1]) - E^(12*C[1])*x^6])^(1/3) + (-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) -
 E^(12*C[1])*x^6])^(1/3)]/Sqrt[2]}, {y[x] -> -Sqrt[(I*(I + Sqrt[3])*x^4 - 2*x^2*
(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3) + (-1 - I*S
qrt[3])*(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(2/3))/(-
2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3)]/2}, {y[x] ->
 Sqrt[(I*(I + Sqrt[3])*x^4 - 2*x^2*(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - 
E^(12*C[1])*x^6])^(1/3) + (-1 - I*Sqrt[3])*(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*
C[1]) - E^(12*C[1])*x^6])^(2/3))/(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^
(12*C[1])*x^6])^(1/3)]/2}, {y[x] -> -Sqrt[((-1 - I*Sqrt[3])*x^4 - 2*x^2*(-2*E^(1
2*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3) + I*(I + Sqrt[3])*(
-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(2/3))/(-2*E^(12*C
[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3)]/2}, {y[x] -> Sqrt[((-
1 - I*Sqrt[3])*x^4 - 2*x^2*(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[
1])*x^6])^(1/3) + I*(I + Sqrt[3])*(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E
^(12*C[1])*x^6])^(2/3))/(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])
*x^6])^(1/3)]/2}}

Maple raw input

dsolve(2*y(x)^3*diff(y(x),x) = x^3-x*y(x)^2, y(x),'implicit')

Maple raw output

-1/6*ln((-x^2+2*y(x)^2)/x^2)-1/3*ln((x^2+y(x)^2)/x^2)-ln(x)-_C1 = 0