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 = 0.0425038 (sec), leaf count = 156
\[\left \{\left \{y(x)\to -\sqrt {-\sqrt {e^{4 c_1}+x^4}-x^2}\right \},\left \{y(x)\to \sqrt {-\sqrt {e^{4 c_1}+x^4}-x^2}\right \},\left \{y(x)\to -\sqrt {\sqrt {e^{4 c_1}+x^4}-x^2}\right \},\left \{y(x)\to \sqrt {\sqrt {e^{4 c_1}+x^4}-x^2}\right \},\left \{y(x)\to -x \sqrt {c_1+2 \log (x)}\right \},\left \{y(x)\to x \sqrt {c_1+2 \log (x)}\right \}\right \}\]
Maple ✓
cpu = 0.022 (sec), leaf count = 55
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{2}- \left ( 2\,\ln \left ( x \right ) +{\it \_C1} \right ) {x}^{2}=0,-{\frac {1}{4}\ln \left ( {\frac {2\,{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}} \right ) }-{\frac {1}{2}\ln \left ( {\frac {y \left ( x \right ) }{x}} \right ) }-\ln \left ( x \right ) -{\it \_C1}=0 \right \} \] 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),'implicit')
Maple raw output
-1/4*ln((2*x^2+y(x)^2)/x^2)-1/2*ln(y(x)/x)-ln(x)-_C1 = 0, y(x)^2-(2*ln(x)+_C1)*x^2 = 0