4.19.7 \(-4 a+x^2 y'(x)^2+2 x (y(x)+2 x) y'(x)+y(x)^2=0\)

ODE
\[ -4 a+x^2 y'(x)^2+2 x (y(x)+2 x) y'(x)+y(x)^2=0 \] ODE Classification

[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

Book solution method
Change of variable

Mathematica
cpu = 0.498384 (sec), leaf count = 22

\[\left \{\left \{y(x)\to \frac {c_1 \left (c_1-2 x\right )-a}{x}\right \}\right \}\]

Maple
cpu = 0.231 (sec), leaf count = 34

\[ \left \{ y \relax (x ) ={\frac {-{x}^{2}-a}{x}},y \relax (x ) ={\it \_C1}+{\frac {{{\it \_C1}}^{2}-4\,a}{4\,x}} \right \} \] Mathematica raw input

DSolve[-4*a + y[x]^2 + 2*x*(2*x + y[x])*y'[x] + x^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> (-a + C[1]*(-2*x + C[1]))/x}}

Maple raw input

dsolve(x^2*diff(y(x),x)^2+2*x*(2*x+y(x))*diff(y(x),x)-4*a+y(x)^2 = 0, y(x),'implicit')

Maple raw output

y(x) = (-x^2-a)/x, y(x) = _C1+1/4*(_C1^2-4*a)/x