ODE
\[ \left (x^3-y(x)^4\right ) y'(x)=3 x^2 y(x) \] ODE Classification
[[_homogeneous, `class G`], _rational]
Book solution method
Exact equation, integrating factor
Mathematica ✓
cpu = 0.0724627 (sec), leaf count = 1021
\[\left \{\left \{y(x)\to \frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}{\sqrt [3]{2}}}-\frac {1}{2} \sqrt {-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}-\frac {\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}{\sqrt [3]{2}}-\frac {6 c_1}{\sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}{\sqrt [3]{2}}}}}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}{\sqrt [3]{2}}}+\frac {1}{2} \sqrt {-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}-\frac {\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}{\sqrt [3]{2}}-\frac {6 c_1}{\sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}{\sqrt [3]{2}}}}}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}{\sqrt [3]{2}}}-\frac {1}{2} \sqrt {-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}-\frac {\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}{\sqrt [3]{2}}+\frac {6 c_1}{\sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}{\sqrt [3]{2}}}}}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}-\frac {\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}{\sqrt [3]{2}}+\frac {6 c_1}{\sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}{\sqrt [3]{2}}}}}-\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1^2-\sqrt {81 c_1^4-256 x^9}}}{\sqrt [3]{2}}}\right \}\right \}\]
Maple ✓
cpu = 0.019 (sec), leaf count = 35
\[ \left \{ \ln \left ( x \right ) -{\it \_C1}+{\frac {4}{9}\ln \left ( {\frac { \left ( y \left ( x \right ) \right ) ^{4}+3\,{x}^{3}}{{x}^{3}}} \right ) }-{\frac {4}{9}\ln \left ( {y \left ( x \right ) {x}^{-{\frac {3}{4}}}} \right ) }=0 \right \} \] Mathematica raw input
DSolve[(x^3 - y[x]^4)*y'[x] == 3*x^2*y[x],y[x],x]
Mathematica raw output
{{y[x] -> Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) + (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)]/2 - Sqrt[(-4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) - (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3) - (6*C[1])/Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) + (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)]]/2}, {y[x] -> Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) + (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)]/2 + Sqrt[(-4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) - (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3) - (6*C[1])/Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) + (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)]]/2}, {y[x] -> -Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) + (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)]/2 - Sqrt[(-4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) - (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3) + (6*C[1])/Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) + (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)]]/2}, {y[x] -> -Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) + (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)]/2 + Sqrt[(-4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) - (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3) + (6*C[1])/Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) + (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)]]/2}}
Maple raw input
dsolve((x^3-y(x)^4)*diff(y(x),x) = 3*x^2*y(x), y(x),'implicit')
Maple raw output
ln(x)-_C1+4/9*ln((y(x)^4+3*x^3)/x^3)-4/9*ln(y(x)/x^(3/4)) = 0