4.19.5 $-(a+2xy(x))y^{\prime }(x)+x^2{y}^{\prime }(x{)}^2+y(x{)}^2=0$

ODE
$$-(a+2xy(x))y^{\prime }(x)+x^2{y}^{\prime }(x{)}^2+y(x{)}^2=0$$ ODE Classification

[[_homogeneous, `class G`], _rational, _Clairaut]

Book solution method
Clairaut’s equation and related types, $f(y-x{y}^{\prime },y^{\prime })=0$

Mathematica ✓
cpu = 0.273771 (sec), leaf count = 47

$$\{\{y(x)\to \frac{x-2\sqrt{a}{c}_1}{4c_1^2}\},\{y(x)\to \frac{2\sqrt{a}{c}_1+x}{4c_1^2}\}\}$$

Maple ✓
cpu = 0.044 (sec), leaf count = 36

$$\{y\left(x\right)=-\frac{a}{4x},y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}x-\sqrt{a\mathit{\_}\mathit{C}\mathit{1}},y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}x+\sqrt{a\mathit{\_}\mathit{C}\mathit{1}}\}$$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (x - 2*Sqrt[a]*C[1])/(4*C[1]^2)}, {y[x] -> (x + 2*Sqrt[a]*C[1])/(4*C[1
]^2)}}

Maple raw input

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

Maple raw output

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