##### 4.14.2 $$x \left (x^2-2 y(x)^2\right ) y'(x)=y(x) \left (2 x^2-y(x)^2\right )$$

ODE
$x \left (x^2-2 y(x)^2\right ) y'(x)=y(x) \left (2 x^2-y(x)^2\right )$ ODE Classiﬁcation

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

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.693756 (sec), leaf count = 873

$\left \{\left \{y(x)\to -\sqrt {\frac {\sqrt [3]{\frac {2}{3}} e^{2 c_1} x^2}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}-x^2+\frac {\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}{\sqrt [3]{2} 3^{2/3}}}\right \},\left \{y(x)\to \sqrt {\frac {\sqrt [3]{\frac {2}{3}} e^{2 c_1} x^2}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}-x^2+\frac {\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}{\sqrt [3]{2} 3^{2/3}}}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {-\frac {2 \sqrt [3]{2} \left (-3 i+\sqrt {3}\right ) e^{2 c_1} x^2}{3^{5/6} \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}-4 x^2+\left (\frac {2}{3}\right )^{2/3} \left (-1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {-\frac {2 \sqrt [3]{2} \left (-3 i+\sqrt {3}\right ) e^{2 c_1} x^2}{3^{5/6} \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}-4 x^2+\left (\frac {2}{3}\right )^{2/3} \left (-1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {-\frac {2 \sqrt [3]{2} \left (3 i+\sqrt {3}\right ) e^{2 c_1} x^2}{3^{5/6} \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}-4 x^2+i \left (\frac {2}{3}\right )^{2/3} \left (i+\sqrt {3}\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {-\frac {2 \sqrt [3]{2} \left (3 i+\sqrt {3}\right ) e^{2 c_1} x^2}{3^{5/6} \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}-4 x^2+i \left (\frac {2}{3}\right )^{2/3} \left (i+\sqrt {3}\right ) \sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}\right \}\right \}$

Maple
cpu = 0.442 (sec), leaf count = 807

$\left [y \left (x \right ) = -\frac {216 x^{2} \textit {\_C1}}{\left (6 \left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}+\frac {24}{\left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}+12\right )^{\frac {3}{2}}}, y \left (x \right ) = \frac {216 x^{2} \textit {\_C1}}{\left (6 \left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}+\frac {24}{\left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}+12\right )^{\frac {3}{2}}}, y \left (x \right ) = -\frac {216 x^{2} \textit {\_C1}}{\left (-3 \left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}-\frac {12}{\left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}+12-18 i \sqrt {3}\, \left (\frac {\left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}\right )\right )^{\frac {3}{2}}}, y \left (x \right ) = \frac {216 x^{2} \textit {\_C1}}{\left (-3 \left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}-\frac {12}{\left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}+12-18 i \sqrt {3}\, \left (\frac {\left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}\right )\right )^{\frac {3}{2}}}, y \left (x \right ) = -\frac {216 x^{2} \textit {\_C1}}{\left (-3 \left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}-\frac {12}{\left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}+12+18 i \sqrt {3}\, \left (\frac {\left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}\right )\right )^{\frac {3}{2}}}, y \left (x \right ) = \frac {216 x^{2} \textit {\_C1}}{\left (-3 \left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}-\frac {12}{\left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}+12+18 i \sqrt {3}\, \left (\frac {\left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 \textit {\_C1}^{2} x^{2}+12 \sqrt {81 x^{4} \textit {\_C1}^{4}-12 \textit {\_C1}^{2} x^{2}}\right )^{\frac {1}{3}}}\right )\right )^{\frac {3}{2}}}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -Sqrt[-x^2 + ((2/3)^(1/3)*E^(2*C[1])*x^2)/(-9*E^(2*C[1])*x^4 + Sqrt[-1
2*E^(6*C[1])*x^6 + 81*E^(4*C[1])*x^8])^(1/3) + (-9*E^(2*C[1])*x^4 + Sqrt[-12*E^(
6*C[1])*x^6 + 81*E^(4*C[1])*x^8])^(1/3)/(2^(1/3)*3^(2/3))]}, {y[x] -> Sqrt[-x^2
+ ((2/3)^(1/3)*E^(2*C[1])*x^2)/(-9*E^(2*C[1])*x^4 + Sqrt[-12*E^(6*C[1])*x^6 + 81
*E^(4*C[1])*x^8])^(1/3) + (-9*E^(2*C[1])*x^4 + Sqrt[-12*E^(6*C[1])*x^6 + 81*E^(4
*C[1])*x^8])^(1/3)/(2^(1/3)*3^(2/3))]}, {y[x] -> -1/2*Sqrt[-4*x^2 - (2*2^(1/3)*(
-3*I + Sqrt[3])*E^(2*C[1])*x^2)/(3^(5/6)*(-9*E^(2*C[1])*x^4 + Sqrt[-12*E^(6*C[1]
)*x^6 + 81*E^(4*C[1])*x^8])^(1/3)) + (2/3)^(2/3)*(-1 - I*Sqrt[3])*(-9*E^(2*C[1])
*x^4 + Sqrt[-12*E^(6*C[1])*x^6 + 81*E^(4*C[1])*x^8])^(1/3)]}, {y[x] -> Sqrt[-4*x
^2 - (2*2^(1/3)*(-3*I + Sqrt[3])*E^(2*C[1])*x^2)/(3^(5/6)*(-9*E^(2*C[1])*x^4 + S
qrt[-12*E^(6*C[1])*x^6 + 81*E^(4*C[1])*x^8])^(1/3)) + (2/3)^(2/3)*(-1 - I*Sqrt[3
])*(-9*E^(2*C[1])*x^4 + Sqrt[-12*E^(6*C[1])*x^6 + 81*E^(4*C[1])*x^8])^(1/3)]/2},
 {y[x] -> -1/2*Sqrt[-4*x^2 - (2*2^(1/3)*(3*I + Sqrt[3])*E^(2*C[1])*x^2)/(3^(5/6)
*(-9*E^(2*C[1])*x^4 + Sqrt[-12*E^(6*C[1])*x^6 + 81*E^(4*C[1])*x^8])^(1/3)) + I*(
2/3)^(2/3)*(I + Sqrt[3])*(-9*E^(2*C[1])*x^4 + Sqrt[-12*E^(6*C[1])*x^6 + 81*E^(4*
C[1])*x^8])^(1/3)]}, {y[x] -> Sqrt[-4*x^2 - (2*2^(1/3)*(3*I + Sqrt[3])*E^(2*C[1]
)*x^2)/(3^(5/6)*(-9*E^(2*C[1])*x^4 + Sqrt[-12*E^(6*C[1])*x^6 + 81*E^(4*C[1])*x^8
])^(1/3)) + I*(2/3)^(2/3)*(I + Sqrt[3])*(-9*E^(2*C[1])*x^4 + Sqrt[-12*E^(6*C[1])
*x^6 + 81*E^(4*C[1])*x^8])^(1/3)]/2}}

Maple raw input

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

Maple raw output

[y(x) = -216*x^2*_C1/(6*(8-108*_C1^2*x^2+12*(81*_C1^4*x^4-12*_C1^2*x^2)^(1/2))^(
1/3)+24/(8-108*_C1^2*x^2+12*(81*_C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/3)+12)^(3/2),
y(x) = 216*x^2*_C1/(6*(8-108*_C1^2*x^2+12*(81*_C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/
3)+24/(8-108*_C1^2*x^2+12*(81*_C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/3)+12)^(3/2), y(
x) = -216*x^2*_C1/(-3*(8-108*_C1^2*x^2+12*(81*_C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/
3)-12/(8-108*_C1^2*x^2+12*(81*_C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/3)+12-18*I*3^(1/
2)*(1/6*(8-108*_C1^2*x^2+12*(81*_C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/3)-2/3/(8-108*
_C1^2*x^2+12*(81*_C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/3)))^(3/2), y(x) = 216*x^2*_C
1/(-3*(8-108*_C1^2*x^2+12*(81*_C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/3)-12/(8-108*_C1
^2*x^2+12*(81*_C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/3)+12-18*I*3^(1/2)*(1/6*(8-108*_
C1^2*x^2+12*(81*_C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/3)-2/3/(8-108*_C1^2*x^2+12*(81
*_C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/3)))^(3/2), y(x) = -216*x^2*_C1/(-3*(8-108*_C
1^2*x^2+12*(81*_C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/3)-12/(8-108*_C1^2*x^2+12*(81*_
C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/3)+12+18*I*3^(1/2)*(1/6*(8-108*_C1^2*x^2+12*(81
*_C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/3)-2/3/(8-108*_C1^2*x^2+12*(81*_C1^4*x^4-12*_
C1^2*x^2)^(1/2))^(1/3)))^(3/2), y(x) = 216*x^2*_C1/(-3*(8-108*_C1^2*x^2+12*(81*_
C1^4*x^4-12*_C1^2*x^2)^(1/2))^(1/3)-12/(8-108*_C1^2*x^2+12*(81*_C1^4*x^4-12*_C1^
2*x^2)^(1/2))^(1/3)+12+18*I*3^(1/2)*(1/6*(8-108*_C1^2*x^2+12*(81*_C1^4*x^4-12*_C
1^2*x^2)^(1/2))^(1/3)-2/3/(8-108*_C1^2*x^2+12*(81*_C1^4*x^4-12*_C1^2*x^2)^(1/2))
^(1/3)))^(3/2)]