##### 4.19.29 $$a x^2 y'(x)^2+(1-a) a x^2-2 a x y(x) y'(x)+y(x)^2=0$$

ODE
$a x^2 y'(x)^2+(1-a) a x^2-2 a x y(x) y'(x)+y(x)^2=0$ ODE Classiﬁcation

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

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

Mathematica
cpu = 0.590694 (sec), leaf count = 241

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

Maple
cpu = 0.265 (sec), leaf count = 138

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

DSolve[(1 - a)*a*x^2 + y[x]^2 - 2*a*x*y[x]*y'[x] + a*x^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(a*x^2*diff(y(x),x)^2-2*a*x*y(x)*diff(y(x),x)+a*(1-a)*x^2+y(x)^2 = 0, y(x))

Maple raw output

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