4.14.50 \(x \left (x^4-2 y(x)^3\right ) y'(x)+y(x) \left (2 x^4+y(x)^3\right )=0\)

ODE
\[ x \left (x^4-2 y(x)^3\right ) y'(x)+y(x) \left (2 x^4+y(x)^3\right )=0 \] ODE Classification

[[_homogeneous, `class G`], _rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0962784 (sec), leaf count = 1139

\[\left \{\left \{y(x)\to \frac {1}{6} \left (-\sqrt [6]{2} 3^{2/3} \sqrt {\frac {4 \sqrt [3]{6} c_1 x^2+\left (9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}}}}-3 \sqrt {-\frac {4 \sqrt {3} x^4}{\sqrt {\frac {4\ 6^{2/3} c_1 x^2+\sqrt [3]{6} \left (9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}}}}}-\frac {4\ 2^{2/3} c_1 x^2}{\sqrt [3]{27 x^8-3 \sqrt {81 x^{16}-384 x^6 c_1^3}}}-\frac {\sqrt [3]{18 x^8-2 \sqrt {81 x^{16}-384 x^6 c_1^3}}}{3^{2/3}}}\right )\right \},\left \{y(x)\to \frac {1}{6} \left (3 \sqrt {-\frac {4 \sqrt {3} x^4}{\sqrt {\frac {4\ 6^{2/3} c_1 x^2+\sqrt [3]{6} \left (9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}}}}}-\frac {4\ 2^{2/3} c_1 x^2}{\sqrt [3]{27 x^8-3 \sqrt {81 x^{16}-384 x^6 c_1^3}}}-\frac {\sqrt [3]{18 x^8-2 \sqrt {81 x^{16}-384 x^6 c_1^3}}}{3^{2/3}}}-\sqrt [6]{2} 3^{2/3} \sqrt {\frac {4 \sqrt [3]{6} c_1 x^2+\left (9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}}}}\right )\right \},\left \{y(x)\to \frac {1}{6} \left (\sqrt [6]{2} 3^{2/3} \sqrt {\frac {4 \sqrt [3]{6} c_1 x^2+\left (9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}}}}-3 \sqrt {\frac {4 \sqrt {3} x^4}{\sqrt {\frac {4\ 6^{2/3} c_1 x^2+\sqrt [3]{6} \left (9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}}}}}-\frac {4\ 2^{2/3} c_1 x^2}{\sqrt [3]{27 x^8-3 \sqrt {81 x^{16}-384 x^6 c_1^3}}}-\frac {\sqrt [3]{18 x^8-2 \sqrt {81 x^{16}-384 x^6 c_1^3}}}{3^{2/3}}}\right )\right \},\left \{y(x)\to \frac {1}{6} \left (\sqrt [6]{2} 3^{2/3} \sqrt {\frac {4 \sqrt [3]{6} c_1 x^2+\left (9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}}}}+3 \sqrt {\frac {4 \sqrt {3} x^4}{\sqrt {\frac {4\ 6^{2/3} c_1 x^2+\sqrt [3]{6} \left (9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}\right ){}^{2/3}}{\sqrt [3]{9 x^8-\sqrt {81 x^{16}-384 x^6 c_1^3}}}}}-\frac {4\ 2^{2/3} c_1 x^2}{\sqrt [3]{27 x^8-3 \sqrt {81 x^{16}-384 x^6 c_1^3}}}-\frac {\sqrt [3]{18 x^8-2 \sqrt {81 x^{16}-384 x^6 c_1^3}}}{3^{2/3}}}\right )\right \}\right \}\]

Maple
cpu = 0.016 (sec), leaf count = 29

\[ \left \{ \ln \relax (x ) -{\it \_C1}+{\frac {3}{10}\ln \left (-2\,{\frac { \left ({x}^{4}-1/2\, \left (y \relax (x ) \right ) ^{3} \right ) y \relax (x ) }{{x}^{16/3}}} \right ) }=0 \right \} \] Mathematica raw input

DSolve[y[x]*(2*x^4 + y[x]^3) + x*(x^4 - 2*y[x]^3)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (-(2^(1/6)*3^(2/3)*Sqrt[(4*6^(1/3)*x^2*C[1] + (9*x^8 - Sqrt[81*x^16 - 
384*x^6*C[1]^3])^(2/3))/(9*x^8 - Sqrt[81*x^16 - 384*x^6*C[1]^3])^(1/3)]) - 3*Sqr
t[(-4*2^(2/3)*x^2*C[1])/(27*x^8 - 3*Sqrt[81*x^16 - 384*x^6*C[1]^3])^(1/3) - (18*
x^8 - 2*Sqrt[81*x^16 - 384*x^6*C[1]^3])^(1/3)/3^(2/3) - (4*Sqrt[3]*x^4)/Sqrt[(4*
6^(2/3)*x^2*C[1] + 6^(1/3)*(9*x^8 - Sqrt[81*x^16 - 384*x^6*C[1]^3])^(2/3))/(9*x^
8 - Sqrt[81*x^16 - 384*x^6*C[1]^3])^(1/3)]])/6}, {y[x] -> (-(2^(1/6)*3^(2/3)*Sqr
t[(4*6^(1/3)*x^2*C[1] + (9*x^8 - Sqrt[81*x^16 - 384*x^6*C[1]^3])^(2/3))/(9*x^8 -
 Sqrt[81*x^16 - 384*x^6*C[1]^3])^(1/3)]) + 3*Sqrt[(-4*2^(2/3)*x^2*C[1])/(27*x^8 
- 3*Sqrt[81*x^16 - 384*x^6*C[1]^3])^(1/3) - (18*x^8 - 2*Sqrt[81*x^16 - 384*x^6*C
[1]^3])^(1/3)/3^(2/3) - (4*Sqrt[3]*x^4)/Sqrt[(4*6^(2/3)*x^2*C[1] + 6^(1/3)*(9*x^
8 - Sqrt[81*x^16 - 384*x^6*C[1]^3])^(2/3))/(9*x^8 - Sqrt[81*x^16 - 384*x^6*C[1]^
3])^(1/3)]])/6}, {y[x] -> (2^(1/6)*3^(2/3)*Sqrt[(4*6^(1/3)*x^2*C[1] + (9*x^8 - S
qrt[81*x^16 - 384*x^6*C[1]^3])^(2/3))/(9*x^8 - Sqrt[81*x^16 - 384*x^6*C[1]^3])^(
1/3)] - 3*Sqrt[(-4*2^(2/3)*x^2*C[1])/(27*x^8 - 3*Sqrt[81*x^16 - 384*x^6*C[1]^3])
^(1/3) - (18*x^8 - 2*Sqrt[81*x^16 - 384*x^6*C[1]^3])^(1/3)/3^(2/3) + (4*Sqrt[3]*
x^4)/Sqrt[(4*6^(2/3)*x^2*C[1] + 6^(1/3)*(9*x^8 - Sqrt[81*x^16 - 384*x^6*C[1]^3])
^(2/3))/(9*x^8 - Sqrt[81*x^16 - 384*x^6*C[1]^3])^(1/3)]])/6}, {y[x] -> (2^(1/6)*
3^(2/3)*Sqrt[(4*6^(1/3)*x^2*C[1] + (9*x^8 - Sqrt[81*x^16 - 384*x^6*C[1]^3])^(2/3
))/(9*x^8 - Sqrt[81*x^16 - 384*x^6*C[1]^3])^(1/3)] + 3*Sqrt[(-4*2^(2/3)*x^2*C[1]
)/(27*x^8 - 3*Sqrt[81*x^16 - 384*x^6*C[1]^3])^(1/3) - (18*x^8 - 2*Sqrt[81*x^16 -
 384*x^6*C[1]^3])^(1/3)/3^(2/3) + (4*Sqrt[3]*x^4)/Sqrt[(4*6^(2/3)*x^2*C[1] + 6^(
1/3)*(9*x^8 - Sqrt[81*x^16 - 384*x^6*C[1]^3])^(2/3))/(9*x^8 - Sqrt[81*x^16 - 384
*x^6*C[1]^3])^(1/3)]])/6}}

Maple raw input

dsolve(x*(x^4-2*y(x)^3)*diff(y(x),x)+(2*x^4+y(x)^3)*y(x) = 0, y(x),'implicit')

Maple raw output

ln(x)-_C1+3/10*ln(-2*(x^4-1/2*y(x)^3)/x^(16/3)*y(x)) = 0