ODE
\[ a+y(x)^2 y'(x)^2+2 x y(x) y'(x)-y(x)^2=0 \] ODE Classification
[_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Book solution method
Change of variable
Mathematica ✓
cpu = 0.461299 (sec), leaf count = 41
\[\left \{\left \{y(x)\to -\sqrt {a+c_1 \left (c_1-2 x\right )}\right \},\left \{y(x)\to \sqrt {a+c_1 \left (c_1-2 x\right )}\right \}\right \}\]
Maple ✓
cpu = 6. (sec), leaf count = 522
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}-a=0,\int _{{\it \_b}}^{x}\!{1 \left ( {\it \_a}-\sqrt {{{\it \_a}}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}-a} \right ) \left ( \left ( 2\,{{\it \_a}}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}-2\,a \right ) \sqrt {{{\it \_a}}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}-a}+2\,{\it \_a}\, \left ( -{{\it \_a}}^{2}- \left ( y \left ( x \right ) \right ) ^{2}+a \right ) \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!-{{\it \_f} \left ( -2\,\sqrt {{{\it \_f}}^{2}+{x}^{2}-a}{x}^{2}-\sqrt {{{\it \_f}}^{2}+{x}^{2}-a}{{\it \_f}}^{2}+2\,{x}^{3}+2\,x{{\it \_f}}^{2}+2\,\sqrt {{{\it \_f}}^{2}+{x}^{2}-a}a-2\,ax \right ) ^{-1}}-\int _{{\it \_b}}^{x}\!{\frac {1}{4} \left ( {{{\it \_f}}^{3}{\it \_a}{\frac {1}{\sqrt {{{\it \_a}}^{2}+{{\it \_f}}^{2}-a}}}}+4\, \left ( {\it \_a}-1/2\,\sqrt {{{\it \_a}}^{2}+{{\it \_f}}^{2}-a} \right ) {\it \_f}\, \left ( {\it \_a}-\sqrt {{{\it \_a}}^{2}+{{\it \_f}}^{2}-a} \right ) \right ) \left ( \left ( {{\it \_a}}^{2}+{\frac {{{\it \_f}}^{2}}{2}}-a \right ) \sqrt {{{\it \_a}}^{2}+{{\it \_f}}^{2}-a}+{\it \_a}\, \left ( -{{\it \_a}}^{2}-{{\it \_f}}^{2}+a \right ) \right ) ^{-2}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0,\int _{{\it \_b}}^{x}\!{1 \left ( {\it \_a}+\sqrt {{{\it \_a}}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}-a} \right ) \left ( \left ( -2\,{{\it \_a}}^{2}- \left ( y \left ( x \right ) \right ) ^{2}+2\,a \right ) \sqrt {{{\it \_a}}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}-a}+2\,{\it \_a}\, \left ( -{{\it \_a}}^{2}- \left ( y \left ( x \right ) \right ) ^{2}+a \right ) \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!{{\it \_f} \left ( -2\,\sqrt {{{\it \_f}}^{2}+{x}^{2}-a}{x}^{2}-\sqrt {{{\it \_f}}^{2}+{x}^{2}-a}{{\it \_f}}^{2}-2\,{x}^{3}-2\,x{{\it \_f}}^{2}+2\,\sqrt {{{\it \_f}}^{2}+{x}^{2}-a}a+2\,ax \right ) ^{-1}}-\int _{{\it \_b}}^{x}\!{\frac {1}{4} \left ( -{{{\it \_f}}^{3}{\it \_a}{\frac {1}{\sqrt {{{\it \_a}}^{2}+{{\it \_f}}^{2}-a}}}}+4\, \left ( {\it \_a}+1/2\,\sqrt {{{\it \_a}}^{2}+{{\it \_f}}^{2}-a} \right ) \left ( {\it \_a}+\sqrt {{{\it \_a}}^{2}+{{\it \_f}}^{2}-a} \right ) {\it \_f} \right ) \left ( \left ( -{{\it \_a}}^{2}-{\frac {{{\it \_f}}^{2}}{2}}+a \right ) \sqrt {{{\it \_a}}^{2}+{{\it \_f}}^{2}-a}+{\it \_a}\, \left ( -{{\it \_a}}^{2}-{{\it \_f}}^{2}+a \right ) \right ) ^{-2}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[a - y[x]^2 + 2*x*y[x]*y'[x] + y[x]^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -Sqrt[a + C[1]*(-2*x + C[1])]}, {y[x] -> Sqrt[a + C[1]*(-2*x + C[1])]}}
Maple raw input
dsolve(y(x)^2*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+a-y(x)^2 = 0, y(x),'implicit')
Maple raw output
y(x)^2+x^2-a = 0, Int((_a+(_a^2+y(x)^2-a)^(1/2))/((-2*_a^2-y(x)^2+2*a)*(_a^2+y(x)^2-a)^(1/2)+2*_a*(-_a^2-y(x)^2+a)),_a = _b .. x)+Intat(_f/(-2*(_f^2+x^2-a)^(1/2)*x^2-(_f^2+x^2-a)^(1/2)*_f^2-2*x^3-2*x*_f^2+2*(_f^2+x^2-a)^(1/2)*a+2*a*x)-Int(1/4*(-_f^3*_a/(_a^2+_f^2-a)^(1/2)+4*(_a+1/2*(_a^2+_f^2-a)^(1/2))*(_a+(_a^2+_f^2-a)^(1/2))*_f)/((-_a^2-1/2*_f^2+a)*(_a^2+_f^2-a)^(1/2)+_a*(-_a^2-_f^2+a))^2,_a = _b .. x),_f = y(x))+_C1 = 0, Int((_a-(_a^2+y(x)^2-a)^(1/2))/((2*_a^2+y(x)^2-2*a)*(_a^2+y(x)^2-a)^(1/2)+2*_a*(-_a^2-y(x)^2+a)),_a = _b .. x)+Intat(-_f/(-2*(_f^2+x^2-a)^(1/2)*x^2-(_f^2+x^2-a)^(1/2)*_f^2+2*x^3+2*x*_f^2+2*(_f^2+x^2-a)^(1/2)*a-2*a*x)-Int(1/4*(_f^3*_a/(_a^2+_f^2-a)^(1/2)+4*(_a-1/2*(_a^2+_f^2-a)^(1/2))*_f*(_a-(_a^2+_f^2-a)^(1/2)))/((_a^2+1/2*_f^2-a)*(_a^2+_f^2-a)^(1/2)+_a*(-_a^2-_f^2+a))^2,_a = _b .. x),_f = y(x))+_C1 = 0