ODE
\[ x y(x) \left (x^2+y(x)^2\right ) y'(x)^2-x y(x) \left (x^2+y(x)^2\right )-\left (x^4+x^2 y(x)^2+y(x)^4\right ) y'(x)=0 \] ODE Classification
[[_homogeneous, `class A`], _rational, _Bernoulli]
Book solution method
Homogeneous ODE, \(x^n f\left ( \frac {y}{x} , y' \right )=0\), Solve for \(p\)
Mathematica ✓
cpu = 0.395528 (sec), leaf count = 156
\[\left \{\left \{y(x)\to -\sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}}\right \},\left \{y(x)\to \sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}}\right \},\left \{y(x)\to -\sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}}\right \},\left \{y(x)\to \sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}}\right \},\left \{y(x)\to -x \sqrt {2 \log (x)+c_1}\right \},\left \{y(x)\to x \sqrt {2 \log (x)+c_1}\right \}\right \}\]
Maple ✓
cpu = 1.308 (sec), leaf count = 250
\[\left [y \left (x \right ) = \frac {\sqrt {\left (x^{2} \textit {\_C1} -\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1} \,x^{2}}}{x \left (x^{2} \textit {\_C1} -\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1}}, y \left (x \right ) = \frac {\sqrt {\left (x^{2} \textit {\_C1} +\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1} \,x^{2}}}{x \left (x^{2} \textit {\_C1} +\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1}}, y \left (x \right ) = -\frac {\sqrt {\left (x^{2} \textit {\_C1} -\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1} \,x^{2}}}{x \left (x^{2} \textit {\_C1} -\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1}}, y \left (x \right ) = -\frac {\sqrt {\left (x^{2} \textit {\_C1} +\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1} \,x^{2}}}{x \left (x^{2} \textit {\_C1} +\sqrt {x^{4} \textit {\_C1}^{2}+1}\right ) \textit {\_C1}}, y \left (x \right ) = \sqrt {2 \ln \left (x \right )+\textit {\_C1}}\, x, y \left (x \right ) = -\sqrt {2 \ln \left (x \right )+\textit {\_C1}}\, x\right ]\] Mathematica raw input
DSolve[-(x*y[x]*(x^2 + y[x]^2)) - (x^4 + x^2*y[x]^2 + y[x]^4)*y'[x] + x*y[x]*(x^2 + y[x]^2)*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -Sqrt[-x^2 - Sqrt[E^(4*C[1]) + x^4]]}, {y[x] -> Sqrt[-x^2 - Sqrt[E^(4*C[1]) + x^4]]}, {y[x] -> -Sqrt[-x^2 + Sqrt[E^(4*C[1]) + x^4]]}, {y[x] -> Sqrt[-x^2 + Sqrt[E^(4*C[1]) + x^4]]}, {y[x] -> -(x*Sqrt[C[1] + 2*Log[x]])}, {y[x] -> x*Sqrt[C[1] + 2*Log[x]]}}
Maple raw input
dsolve(x*y(x)*(x^2+y(x)^2)*diff(y(x),x)^2-(x^4+x^2*y(x)^2+y(x)^4)*diff(y(x),x)-x*y(x)*(x^2+y(x)^2) = 0, y(x))
Maple raw output
[y(x) = 1/x/(x^2*_C1-(_C1^2*x^4+1)^(1/2))/_C1*((x^2*_C1-(_C1^2*x^4+1)^(1/2))*_C1*x^2)^(1/2), y(x) = 1/x/(x^2*_C1+(_C1^2*x^4+1)^(1/2))/_C1*((x^2*_C1+(_C1^2*x^4+1)^(1/2))*_C1*x^2)^(1/2), y(x) = -1/x/(x^2*_C1-(_C1^2*x^4+1)^(1/2))/_C1*((x^2*_C1-(_C1^2*x^4+1)^(1/2))*_C1*x^2)^(1/2), y(x) = -1/x/(x^2*_C1+(_C1^2*x^4+1)^(1/2))/_C1*((x^2*_C1+(_C1^2*x^4+1)^(1/2))*_C1*x^2)^(1/2), y(x) = (2*ln(x)+_C1)^(1/2)*x, y(x) = -(2*ln(x)+_C1)^(1/2)*x]