ODE
\[ \left (x^2-y(x)^2\right ) y'(x)+x y(x) y'(x)^2-x y(x)=0 \] ODE Classification
[_separable]
Book solution method
No Missing Variables ODE, Solve for \(y'\)
Mathematica ✓
cpu = 0.00653316 (sec), leaf count = 47
\[\left \{\left \{y(x)\to c_1 x\right \},\left \{y(x)\to -\sqrt {2 c_1-x^2}\right \},\left \{y(x)\to \sqrt {2 c_1-x^2}\right \}\right \}\]
Maple ✓
cpu = 0.01 (sec), leaf count = 20
\[ \left \{ {x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}-{\it \_C1}=0,y \left ( x \right ) ={\it \_C1}\,x \right \} \] Mathematica raw input
DSolve[-(x*y[x]) + (x^2 - y[x]^2)*y'[x] + x*y[x]*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> x*C[1]}, {y[x] -> -Sqrt[-x^2 + 2*C[1]]}, {y[x] -> Sqrt[-x^2 + 2*C[1]]}}
Maple raw input
dsolve(x*y(x)*diff(y(x),x)^2+(x^2-y(x)^2)*diff(y(x),x)-x*y(x) = 0, y(x),'implicit')
Maple raw output
x^2+y(x)^2-_C1 = 0, y(x) = _C1*x