ODE
\[ x \left (x^2+2 y(x)^2\right ) y'(x)=y(x) \left (2 x^2+3 y(x)^2\right ) \] ODE Classification
[[_homogeneous, `class A`], _rational, _dAlembert]
Book solution method
Homogeneous equation
Mathematica ✓
cpu = 0.043949 (sec), leaf count = 163
\[\left \{\left \{y(x)\to -\frac {\sqrt {-\sqrt {4 e^{2 c_1} x^6+x^4}-x^2}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {-\sqrt {4 e^{2 c_1} x^6+x^4}-x^2}}{\sqrt {2}}\right \},\left \{y(x)\to -\frac {\sqrt {\sqrt {4 e^{2 c_1} x^6+x^4}-x^2}}{\sqrt {2}}\right \},\left \{y(x)\to \sqrt {\frac {1}{2} \sqrt {4 e^{2 c_1} x^6+x^4}-\frac {x^2}{2}}\right \}\right \}\]
Maple ✓
cpu = 0.019 (sec), leaf count = 33
\[ \left \{ {\frac {1}{2}\ln \left ( {\frac {{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}} \right ) }+\ln \left ( {\frac {y \left ( x \right ) }{x}} \right ) -\ln \left ( x \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[x*(x^2 + 2*y[x]^2)*y'[x] == y[x]*(2*x^2 + 3*y[x]^2),y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[-x^2 - Sqrt[x^4 + 4*E^(2*C[1])*x^6]]/Sqrt[2])}, {y[x] -> Sqrt[-x^2 - Sqrt[x^4 + 4*E^(2*C[1])*x^6]]/Sqrt[2]}, {y[x] -> -(Sqrt[-x^2 + Sqrt[x^4 + 4*E^(2*C[1])*x^6]]/Sqrt[2])}, {y[x] -> Sqrt[-x^2/2 + Sqrt[x^4 + 4*E^(2*C[1])*x^6]/2]}}
Maple raw input
dsolve(x*(x^2+2*y(x)^2)*diff(y(x),x) = (2*x^2+3*y(x)^2)*y(x), y(x),'implicit')
Maple raw output
1/2*ln((x^2+y(x)^2)/x^2)+ln(y(x)/x)-ln(x)-_C1 = 0