4.13.37 \(\left (a y(x)^2+x^2+x y(x)\right ) y'(x)=a x^2+x y(x)+y(x)^2\)

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

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0451534 (sec), leaf count = 51

\[\text {Solve}\left [\frac {1}{3} \left ((a-1) \log \left (\frac {x^2+x y(x)+y(x)^2}{x^2}\right )+(a+2) \log \left (1-\frac {y(x)}{x}\right )\right )+a \log (x)=c_1,y(x)\right ]\]

Maple
cpu = 0.023 (sec), leaf count = 56

\[ \left \{ {\frac {1}{3\,a} \left (\left (1-a \right ) \ln \left ({\frac {{x}^{2}+xy \relax (x ) + \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) + \left (-a-2 \right ) \ln \left ({\frac {y \relax (x ) -x}{x}} \right ) -3\,a \left ({\it \_C1}+\ln \relax (x ) \right ) \right ) }=0 \right \} \] Mathematica raw input

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

Mathematica raw output

Solve[a*Log[x] + ((2 + a)*Log[1 - y[x]/x] + (-1 + a)*Log[(x^2 + x*y[x] + y[x]^2)
/x^2])/3 == C[1], y[x]]

Maple raw input

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

Maple raw output

1/3*((1-a)*ln((x^2+x*y(x)+y(x)^2)/x^2)+(-a-2)*ln((y(x)-x)/x)-3*a*(_C1+ln(x)))/a 
= 0