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 = 0.0867309 (sec), leaf count = 109
\[\left \{\left \{y(x)\to -\sqrt {e^{c_1} \left (e^{c_1}-2 \sqrt {a} x\right )}\right \},\left \{y(x)\to \sqrt {e^{c_1} \left (e^{c_1}-2 \sqrt {a} x\right )}\right \},\left \{y(x)\to -\sqrt {e^{c_1} \left (2 \sqrt {a} x+e^{c_1}\right )}\right \},\left \{y(x)\to \sqrt {e^{c_1} \left (2 \sqrt {a} x+e^{c_1}\right )}\right \}\right \}\]
Maple ✓
cpu = 0.08 (sec), leaf count = 107
\[ \left \{ a{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}=0,\ln \left ( x \right ) -\int ^{{\frac {y \left ( x \right ) }{x}}}\!{\frac {1}{{\it \_a}\, \left ( {{\it \_a}}^{2}+a \right ) } \left ( -{{\it \_a}}^{2}+\sqrt {{{\it \_a}}^{2}a+{a}^{2}}-a \right ) }{d{\it \_a}}-{\it \_C1}=0,\ln \left ( x \right ) +\int ^{{\frac {y \left ( x \right ) }{x}}}\!{\frac {1}{{\it \_a}\, \left ( {{\it \_a}}^{2}+a \right ) } \left ( {{\it \_a}}^{2}+\sqrt {{{\it \_a}}^{2}a+{a}^{2}}+a \right ) }{d{\it \_a}}-{\it \_C1}=0 \right \} \] 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)]}, {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),'implicit')
Maple raw output
a*x^2+y(x)^2 = 0, ln(x)-Intat((-_a^2+(_a^2*a+a^2)^(1/2)-a)/_a/(_a^2+a),_a = y(x)/x)-_C1 = 0, ln(x)+Intat((_a^2+(_a^2*a+a^2)^(1/2)+a)/_a/(_a^2+a),_a = y(x)/x)-_C1 = 0