ODE
\[ 2 x y(x) y'(x)=a x+y(x)^2 \] ODE Classification
[[_homogeneous, `class G`], _rational, _Bernoulli]
Book solution method
The Bernoulli ODE
Mathematica ✓
cpu = 0.00788918 (sec), leaf count = 44
\[\left \{\left \{y(x)\to -\sqrt {x} \sqrt {a \log (x)+c_1}\right \},\left \{y(x)\to \sqrt {x} \sqrt {a \log (x)+c_1}\right \}\right \}\]
Maple ✓
cpu = 0.006 (sec), leaf count = 18
\[ \left \{ -\ln \left ( x \right ) ax+ \left ( y \left ( x \right ) \right ) ^{2}-{\it \_C1}\,x=0 \right \} \] Mathematica raw input
DSolve[2*x*y[x]*y'[x] == a*x + y[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[x]*Sqrt[C[1] + a*Log[x]])}, {y[x] -> Sqrt[x]*Sqrt[C[1] + a*Log[x]]}}
Maple raw input
dsolve(2*x*y(x)*diff(y(x),x) = a*x+y(x)^2, y(x),'implicit')
Maple raw output
-ln(x)*a*x+y(x)^2-_C1*x = 0