##### 4.19.48 $$2 a x y'(x)-a y(x)+y(x) y'(x)^2=0$$

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

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

Book solution method
Homogeneous ODE, $$x^n f\left ( \frac {y}{x} , y' \right )=0$$

Mathematica
cpu = 0.318412 (sec), leaf count = 55

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

Maple
cpu = 0.288 (sec), leaf count = 114

$\left [y \left (x \right ) = \sqrt {-a}\, x, y \left (x \right ) = -\sqrt {-a}\, x, y \left (x \right ) = \RootOf \left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {-\textit {\_a}^{2}+\sqrt {a \,\textit {\_a}^{2}+a^{2}}-a}{\textit {\_a} \left (\textit {\_a}^{2}+a \right )}d \textit {\_a} +\textit {\_C1} \right ) x, y \left (x \right ) = \RootOf \left (-\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}+\sqrt {a \,\textit {\_a}^{2}+a^{2}}+a}{\textit {\_a} \left (\textit {\_a}^{2}+a \right )}d \textit {\_a} \right )+\textit {\_C1} \right ) x\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = (-a)^(1/2)*x, y(x) = -(-a)^(1/2)*x, y(x) = RootOf(-ln(x)+Intat((-_a^2+(_
a^2*a+a^2)^(1/2)-a)/_a/(_a^2+a),_a = _Z)+_C1)*x, y(x) = RootOf(-ln(x)-Intat((_a^
2+(_a^2*a+a^2)^(1/2)+a)/_a/(_a^2+a),_a = _Z)+_C1)*x]