4.13.38 \(\left (a x^2-a y(x)^2+2 x y(x)\right ) y'(x)-2 a x y(x)+x^2-y(x)^2=0\)

ODE
\[ \left (a x^2-a y(x)^2+2 x y(x)\right ) y'(x)-2 a x y(x)+x^2-y(x)^2=0 \] ODE Classification

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0575669 (sec), leaf count = 87

\[\left \{\left \{y(x)\to \frac {1}{2} \left (a \left (-e^{c_1}\right )-\sqrt {a^2 e^{2 c_1}+4 x \left (e^{c_1}-x\right )}\right )\right \},\left \{y(x)\to \frac {1}{2} \left (\sqrt {a^2 e^{2 c_1}+4 x \left (e^{c_1}-x\right )}-a e^{c_1}\right )\right \}\right \}\]

Maple
cpu = 0.021 (sec), leaf count = 39

\[ \left \{ \ln \left ({\frac {ay \relax (x ) -x}{x}} \right ) -\ln \left ({\frac {{x}^{2}+ \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) -\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (-(a*E^C[1]) - Sqrt[a^2*E^(2*C[1]) + 4*(E^C[1] - x)*x])/2}, {y[x] -> (
-(a*E^C[1]) + Sqrt[a^2*E^(2*C[1]) + 4*(E^C[1] - x)*x])/2}}

Maple raw input

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

Maple raw output

ln((a*y(x)-x)/x)-ln((x^2+y(x)^2)/x^2)-ln(x)-_C1 = 0