##### 4.13.38 $$\left (a x^2-a y(x)^2+2 x y(x)\right ) y'(x)-2 a x y(x)+x^2-y(x)^2=0$$

ODE
$\left (a x^2-a y(x)^2+2 x y(x)\right ) y'(x)-2 a x y(x)+x^2-y(x)^2=0$ ODE Classiﬁcation

[[_homogeneous, class A], _rational, _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.519991 (sec), leaf count = 87

$\left \{\left \{y(x)\to \frac {1}{2} \left (a \left (-e^{c_1}\right )-\sqrt {a^2 e^{2 c_1}+4 x \left (-x+e^{c_1}\right )}\right )\right \},\left \{y(x)\to \frac {1}{2} \left (\sqrt {a^2 e^{2 c_1}+4 x \left (-x+e^{c_1}\right )}-a e^{c_1}\right )\right \}\right \}$

Maple
cpu = 0.062 (sec), leaf count = 59

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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