4.12.12 $$a x y(x) y'(x)+x^2-y(x)^2=0$$

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

[[_homogeneous, class A], _rational, _Bernoulli]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.256245 (sec), leaf count = 72

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

Maple
cpu = 0.023 (sec), leaf count = 84

$\left [y \left (x \right ) = \frac {\sqrt {\left (a -1\right ) \left (x^{\frac {2}{a}} \textit {\_C1} a -x^{\frac {2}{a}} \textit {\_C1} -x^{2}\right )}}{a -1}, y \left (x \right ) = -\frac {\sqrt {\left (a -1\right ) \left (x^{\frac {2}{a}} \textit {\_C1} a -x^{\frac {2}{a}} \textit {\_C1} -x^{2}\right )}}{a -1}\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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