ODE
\[ \left (x^3-y(x)^3\right ) y'(x)+x^2 y(x)=0 \] ODE Classification
[[_homogeneous, `class A`], _rational, _dAlembert]
Book solution method
Homogeneous equation
Mathematica ✓
cpu = 0.0461361 (sec), leaf count = 201
\[\left \{\left \{y(x)\to \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}}\right \},\left \{y(x)\to -\sqrt [3]{-1} \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}}\right \},\left \{y(x)\to (-1)^{2/3} \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}}\right \},\left \{y(x)\to \sqrt [3]{\sqrt {x^6-e^{6 c_1}}+x^3}\right \},\left \{y(x)\to -\sqrt [3]{-1} \sqrt [3]{\sqrt {x^6-e^{6 c_1}}+x^3}\right \},\left \{y(x)\to (-1)^{2/3} \sqrt [3]{\sqrt {x^6-e^{6 c_1}}+x^3}\right \}\right \}\]
Maple ✓
cpu = 0.019 (sec), leaf count = 37
\[ \left \{ -{\frac {1}{6}\ln \left ( {\frac {-2\,{x}^{3}+ \left ( y \left ( x \right ) \right ) ^{3}}{{x}^{3}}} \right ) }-{\frac {1}{2}\ln \left ( {\frac {y \left ( x \right ) }{x}} \right ) }-\ln \left ( x \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[x^2*y[x] + (x^3 - y[x]^3)*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x^3 - Sqrt[-E^(6*C[1]) + x^6])^(1/3)}, {y[x] -> -((-1)^(1/3)*(x^3 - Sqrt[-E^(6*C[1]) + x^6])^(1/3))}, {y[x] -> (-1)^(2/3)*(x^3 - Sqrt[-E^(6*C[1]) + x^6])^(1/3)}, {y[x] -> (x^3 + Sqrt[-E^(6*C[1]) + x^6])^(1/3)}, {y[x] -> -((-1)^(1/3)*(x^3 + Sqrt[-E^(6*C[1]) + x^6])^(1/3))}, {y[x] -> (-1)^(2/3)*(x^3 + Sqrt[-E^(6*C[1]) + x^6])^(1/3)}}
Maple raw input
dsolve((x^3-y(x)^3)*diff(y(x),x)+x^2*y(x) = 0, y(x),'implicit')
Maple raw output
-1/6*ln((-2*x^3+y(x)^3)/x^3)-1/2*ln(y(x)/x)-ln(x)-_C1 = 0