ODE
\[ \left (x^2-y(x)^2 \left (x^2+y(x)^2\right )\right ) y'(x)^2-2 x y(x) y'(x)+y(x)^2=0 \] ODE Classification
[`y=_G(x,y')`]
Book solution method
Change of variable
Mathematica ✗
cpu = 55.0842 (sec), leaf count = 0 , could not solve
DSolve[y[x]^2 - 2*x*y[x]*Derivative[1][y][x] + (x^2 - y[x]^2*(x^2 + y[x]^2))*Derivative[1][y][x]^2 == 0, y[x], x]
Maple ✓
cpu = 2.323 (sec), leaf count = 32
\[ \left \{ {x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}=0,y \left ( x \right ) +{\it Artanh} \left ( {\frac {1}{\sqrt {{\frac { \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}}+1}}} \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[y[x]^2 - 2*x*y[x]*y'[x] + (x^2 - y[x]^2*(x^2 + y[x]^2))*y'[x]^2 == 0,y[x],x]
Mathematica raw output
DSolve[y[x]^2 - 2*x*y[x]*Derivative[1][y][x] + (x^2 - y[x]^2*(x^2 + y[x]^2))*Derivative[1][y][x]^2 == 0, y[x], x]
Maple raw input
dsolve((x^2-(x^2+y(x)^2)*y(x)^2)*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+y(x)^2 = 0, y(x),'implicit')
Maple raw output
x^2+y(x)^2 = 0, y(x)+arctanh(1/(y(x)^2/x^2+1)^(1/2))-_C1 = 0