ODE
\[ -x \left (x^2-2 y(x)^2\right )-2 y(x)^3 y'(x)+x y(x)^2 y'(x)^2=0 \] ODE Classification
[_separable]
Book solution method
Change of variable
Mathematica ✓
cpu = 0.0116421 (sec), leaf count = 73
\[\left \{\left \{y(x)\to -\sqrt {2 c_1+x^2}\right \},\left \{y(x)\to \sqrt {2 c_1+x^2}\right \},\left \{y(x)\to -\sqrt {c_1 x^4+x^2}\right \},\left \{y(x)\to \sqrt {c_1 x^4+x^2}\right \}\right \}\]
Maple ✓
cpu = 0.013 (sec), leaf count = 34
\[ \left \{ -{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}-{\it \_C1}=0,-{x}^{4}{\it \_C1}-{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}=0 \right \} \] Mathematica raw input
DSolve[-(x*(x^2 - 2*y[x]^2)) - 2*y[x]^3*y'[x] + x*y[x]^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -Sqrt[x^2 + 2*C[1]]}, {y[x] -> Sqrt[x^2 + 2*C[1]]}, {y[x] -> -Sqrt[x^2 + x^4*C[1]]}, {y[x] -> Sqrt[x^2 + x^4*C[1]]}}
Maple raw input
dsolve(x*y(x)^2*diff(y(x),x)^2-2*y(x)^3*diff(y(x),x)-x*(x^2-2*y(x)^2) = 0, y(x),'implicit')
Maple raw output
-x^2+y(x)^2-_C1 = 0, -x^4*_C1-x^2+y(x)^2 = 0