4.15.10 \(\left (x^3-y(x)^4\right ) y'(x)=3 x^2 y(x)\)

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

\[ \left \{ \ln \relax (x ) -{\it \_C1}+{\frac {4}{9}\ln \left ({\frac { \left (y \relax (x ) \right ) ^{4}+3\,{x}^{3}}{{x}^{3}}} \right ) }-{\frac {4}{9}\ln \left ({y \relax (x ) {x}^{-{\frac {3}{4}}}} \right ) }=0 \right \} \] 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 + 8
1*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