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 = 0.50535 (sec), leaf count = 1069

\[\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 {\sqrt {\frac {-2 \sqrt [3]{2} \sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) e^{2 c_1} x^2-12 \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+\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (2 \sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-18 e^{2 c_1} x^4\right ){}^{2/3}}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}}}{2 \sqrt {3}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {-2 \sqrt [3]{2} \sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) e^{2 c_1} x^2-12 \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+\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (2 \sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-18 e^{2 c_1} x^4\right ){}^{2/3}}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}}}{2 \sqrt {3}}\right \},\left \{y(x)\to -\frac {\sqrt {\frac {-2 \sqrt [3]{2} \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) e^{2 c_1} x^2-12 \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+i \sqrt [3]{3} \left (i+\sqrt {3}\right ) \left (2 \sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-18 e^{2 c_1} x^4\right ){}^{2/3}}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}}}{2 \sqrt {3}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {-2 \sqrt [3]{2} \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) e^{2 c_1} x^2-12 \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+i \sqrt [3]{3} \left (i+\sqrt {3}\right ) \left (2 \sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-18 e^{2 c_1} x^4\right ){}^{2/3}}{\sqrt [3]{\sqrt {81 e^{4 c_1} x^8-12 e^{6 c_1} x^6}-9 e^{2 c_1} x^4}}}}{2 \sqrt {3}}\right \}\right \}\]

Maple
cpu = 0.019 (sec), leaf count = 33

\[ \left \{ -{\frac {3}{2}\ln \left ({\frac {{x}^{2}+ \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) }+\ln \left ({\frac {y \relax (x ) }{x}} \right ) -\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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] -> -Sqrt[(-2*2^(1/3)*3^(1/6)*(-3*I
 + Sqrt[3])*E^(2*C[1])*x^2 - 12*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) + 3^(1/3)*(-1 - I*Sqrt[3])*(-18*E^(2*C[1])*x^4 + 2*
Sqrt[-12*E^(6*C[1])*x^6 + 81*E^(4*C[1])*x^8])^(2/3))/(-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*Sqrt[3])}, {y[x] -> Sqrt[(-2*2
^(1/3)*3^(1/6)*(-3*I + Sqrt[3])*E^(2*C[1])*x^2 - 12*x^2*(-9*E^(2*C[1])*x^4 + Sqr
t[-12*E^(6*C[1])*x^6 + 81*E^(4*C[1])*x^8])^(1/3) + 3^(1/3)*(-1 - I*Sqrt[3])*(-18
*E^(2*C[1])*x^4 + 2*Sqrt[-12*E^(6*C[1])*x^6 + 81*E^(4*C[1])*x^8])^(2/3))/(-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*Sqrt[3])},
 {y[x] -> -Sqrt[(-2*2^(1/3)*3^(1/6)*(3*I + Sqrt[3])*E^(2*C[1])*x^2 - 12*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) + I*3^(1/3)
*(I + Sqrt[3])*(-18*E^(2*C[1])*x^4 + 2*Sqrt[-12*E^(6*C[1])*x^6 + 81*E^(4*C[1])*x
^8])^(2/3))/(-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*Sqrt[3])}, {y[x] -> Sqrt[(-2*2^(1/3)*3^(1/6)*(3*I + Sqrt[3])*E^(2*C[1])
*x^2 - 12*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) + I*3^(1/3)*(I + Sqrt[3])*(-18*E^(2*C[1])*x^4 + 2*Sqrt[-12*E^(6*C[1])*x^6
 + 81*E^(4*C[1])*x^8])^(2/3))/(-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*Sqrt[3])}}

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),'implicit')

Maple raw output

-3/2*ln((x^2+y(x)^2)/x^2)+ln(y(x)/x)-ln(x)-_C1 = 0