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 = 0.0867309 (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

\[ \left \{ a{x}^{2}+ \left (y \relax (x ) \right ) ^{2}=0,\ln \relax (x ) -\int ^{{\frac {y \relax (x ) }{x}}}\!{\frac {1}{{\it \_a}\, \left ({{\it \_a}}^{2}+a \right ) } \left (-{{\it \_a}}^{2}+\sqrt {{{\it \_a}}^{2}a+{a}^{2}}-a \right ) }{d{\it \_a}}-{\it \_C1}=0,\ln \relax (x ) +\int ^{{\frac {y \relax (x ) }{x}}}\!{\frac {1}{{\it \_a}\, \left ({{\it \_a}}^{2}+a \right ) } \left ({{\it \_a}}^{2}+\sqrt {{{\it \_a}}^{2}a+{a}^{2}}+a \right ) }{d{\it \_a}}-{\it \_C1}=0 \right \} \] 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)-_C
1 = 0