ODE
\[ -2 x \left (x^2+2 y(x)^2\right ) y(x) y'(x)+\left (x^4+y(x)^2 \left (x^2-y(x)^2\right )\right ) y'(x)^2+y(x)^4=0 \] ODE Classification
[[_homogeneous, `class A`], _dAlembert]
Book solution method
Homogeneous ODE, \(x^n f\left ( \frac {y}{x} , y' \right )=0\), Solve for \(p\)
Mathematica ✓
cpu = 0.188626 (sec), leaf count = 987
\[\left \{\left \{y(x)\to \frac {e^{2 c_1}-3 x^2}{3 \sqrt [3]{-18 e^{c_1} x^2+e^{3 c_1}+3 \sqrt {3} \sqrt {x^6+11 e^{2 c_1} x^4-e^{4 c_1} x^2}}}-\frac {2 e^{c_1}}{3}+\frac {1}{3} \sqrt [3]{-18 e^{c_1} x^2+e^{3 c_1}+3 \sqrt {3} \sqrt {x^6+11 e^{2 c_1} x^4-e^{4 c_1} x^2}}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (3 x^2-e^{2 c_1}\right )}{6 \sqrt [3]{-18 e^{c_1} x^2+e^{3 c_1}+3 \sqrt {3} \sqrt {x^6+11 e^{2 c_1} x^4-e^{4 c_1} x^2}}}-\frac {2 e^{c_1}}{3}+\frac {1}{6} i \left (i+\sqrt {3}\right ) \sqrt [3]{-18 e^{c_1} x^2+e^{3 c_1}+3 \sqrt {3} \sqrt {x^6+11 e^{2 c_1} x^4-e^{4 c_1} x^2}}\right \},\left \{y(x)\to \frac {i \left (i+\sqrt {3}\right ) \left (e^{2 c_1}-3 x^2\right )}{6 \sqrt [3]{-18 e^{c_1} x^2+e^{3 c_1}+3 \sqrt {3} \sqrt {x^6+11 e^{2 c_1} x^4-e^{4 c_1} x^2}}}-\frac {2 e^{c_1}}{3}-\frac {1}{6} \left (1+i \sqrt {3}\right ) \sqrt [3]{-18 e^{c_1} x^2+e^{3 c_1}+3 \sqrt {3} \sqrt {x^6+11 e^{2 c_1} x^4-e^{4 c_1} x^2}}\right \},\left \{y(x)\to \frac {e^{2 c_1}-3 x^2}{3 \sqrt [3]{18 e^{c_1} x^2-e^{3 c_1}+3 \sqrt {3} \sqrt {x^6+11 e^{2 c_1} x^4-e^{4 c_1} x^2}}}+\frac {2 e^{c_1}}{3}+\frac {1}{3} \sqrt [3]{18 e^{c_1} x^2-e^{3 c_1}+3 \sqrt {3} \sqrt {x^6+11 e^{2 c_1} x^4-e^{4 c_1} x^2}}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (3 x^2-e^{2 c_1}\right )}{6 \sqrt [3]{18 e^{c_1} x^2-e^{3 c_1}+3 \sqrt {3} \sqrt {x^6+11 e^{2 c_1} x^4-e^{4 c_1} x^2}}}+\frac {2 e^{c_1}}{3}+\frac {1}{6} i \left (i+\sqrt {3}\right ) \sqrt [3]{18 e^{c_1} x^2-e^{3 c_1}+3 \sqrt {3} \sqrt {x^6+11 e^{2 c_1} x^4-e^{4 c_1} x^2}}\right \},\left \{y(x)\to \frac {i \left (i+\sqrt {3}\right ) \left (e^{2 c_1}-3 x^2\right )}{6 \sqrt [3]{18 e^{c_1} x^2-e^{3 c_1}+3 \sqrt {3} \sqrt {x^6+11 e^{2 c_1} x^4-e^{4 c_1} x^2}}}+\frac {2 e^{c_1}}{3}-\frac {1}{6} \left (1+i \sqrt {3}\right ) \sqrt [3]{18 e^{c_1} x^2-e^{3 c_1}+3 \sqrt {3} \sqrt {x^6+11 e^{2 c_1} x^4-e^{4 c_1} x^2}}\right \}\right \}\]
Maple ✓
cpu = 0.177 (sec), leaf count = 79
\[ \left \{ \ln \left ( x \right ) -{\it Artanh} \left ( {\frac {1}{\sqrt {{\frac {{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}}}} \right ) +{\frac {1}{2}\ln \left ( {\frac {{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}} \right ) }-{\it \_C1}=0,\ln \left ( x \right ) +{\it Artanh} \left ( {\frac {1}{\sqrt {{\frac {{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}}}} \right ) +{\frac {1}{2}\ln \left ( {\frac {{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}} \right ) }-{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[y[x]^4 - 2*x*y[x]*(x^2 + 2*y[x]^2)*y'[x] + (x^4 + y[x]^2*(x^2 - y[x]^2))*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-2*E^C[1])/3 + (E^(2*C[1]) - 3*x^2)/(3*(E^(3*C[1]) - 18*E^C[1]*x^2 + 3*Sqrt[3]*Sqrt[-(E^(4*C[1])*x^2) + 11*E^(2*C[1])*x^4 + x^6])^(1/3)) + (E^(3*C[1]) - 18*E^C[1]*x^2 + 3*Sqrt[3]*Sqrt[-(E^(4*C[1])*x^2) + 11*E^(2*C[1])*x^4 + x^6])^(1/3)/3}, {y[x] -> (-2*E^C[1])/3 + ((1 + I*Sqrt[3])*(-E^(2*C[1]) + 3*x^2))/(6*(E^(3*C[1]) - 18*E^C[1]*x^2 + 3*Sqrt[3]*Sqrt[-(E^(4*C[1])*x^2) + 11*E^(2*C[1])*x^4 + x^6])^(1/3)) + (I/6)*(I + Sqrt[3])*(E^(3*C[1]) - 18*E^C[1]*x^2 + 3*Sqrt[3]*Sqrt[-(E^(4*C[1])*x^2) + 11*E^(2*C[1])*x^4 + x^6])^(1/3)}, {y[x] -> (-2*E^C[1])/3 + ((I/6)*(I + Sqrt[3])*(E^(2*C[1]) - 3*x^2))/(E^(3*C[1]) - 18*E^C[1]*x^2 + 3*Sqrt[3]*Sqrt[-(E^(4*C[1])*x^2) + 11*E^(2*C[1])*x^4 + x^6])^(1/3) - ((1 + I*Sqrt[3])*(E^(3*C[1]) - 18*E^C[1]*x^2 + 3*Sqrt[3]*Sqrt[-(E^(4*C[1])*x^2) + 11*E^(2*C[1])*x^4 + x^6])^(1/3))/6}, {y[x] -> (2*E^C[1])/3 + (E^(2*C[1]) - 3*x^2)/(3*(-E^(3*C[1]) + 18*E^C[1]*x^2 + 3*Sqrt[3]*Sqrt[-(E^(4*C[1])*x^2) + 11*E^(2*C[1])*x^4 + x^6])^(1/3)) + (-E^(3*C[1]) + 18*E^C[1]*x^2 + 3*Sqrt[3]*Sqrt[-(E^(4*C[1])*x^2) + 11*E^(2*C[1])*x^4 + x^6])^(1/3)/3}, {y[x] -> (2*E^C[1])/3 + ((1 + I*Sqrt[3])*(-E^(2*C[1]) + 3*x^2))/(6*(-E^(3*C[1]) + 18*E^C[1]*x^2 + 3*Sqrt[3]*Sqrt[-(E^(4*C[1])*x^2) + 11*E^(2*C[1])*x^4 + x^6])^(1/3)) + (I/6)*(I + Sqrt[3])*(-E^(3*C[1]) + 18*E^C[1]*x^2 + 3*Sqrt[3]*Sqrt[-(E^(4*C[1])*x^2) + 11*E^(2*C[1])*x^4 + x^6])^(1/3)}, {y[x] -> (2*E^C[1])/3 + ((I/6)*(I + Sqrt[3])*(E^(2*C[1]) - 3*x^2))/(-E^(3*C[1]) + 18*E^C[1]*x^2 + 3*Sqrt[3]*Sqrt[-(E^(4*C[1])*x^2) + 11*E^(2*C[1])*x^4 + x^6])^(1/3) - ((1 + I*Sqrt[3])*(-E^(3*C[1]) + 18*E^C[1]*x^2 + 3*Sqrt[3]*Sqrt[-(E^(4*C[1])*x^2) + 11*E^(2*C[1])*x^4 + x^6])^(1/3))/6}}
Maple raw input
dsolve((x^4+(x^2-y(x)^2)*y(x)^2)*diff(y(x),x)^2-2*x*y(x)*(x^2+2*y(x)^2)*diff(y(x),x)+y(x)^4 = 0, y(x),'implicit')
Maple raw output
ln(x)-arctanh(1/((x^2+y(x)^2)/x^2)^(1/2))+1/2*ln((x^2+y(x)^2)/x^2)-_C1 = 0, ln(x)+arctanh(1/((x^2+y(x)^2)/x^2)^(1/2))+1/2*ln((x^2+y(x)^2)/x^2)-_C1 = 0