ODE
\[ x y'(x)+2 y(x)=a x^{2 k} y(x)^k \] ODE Classification
[[_homogeneous, `class G`], _rational, _Bernoulli]
Book solution method
The Bernoulli ODE
Mathematica ✓
cpu = 0.326966 (sec), leaf count = 45
\[\left \{\left \{y(x)\to \left (\frac {1}{2} a x^{2 k}-\frac {1}{2} a k x^{2 k}+c_1 x^{2 k-2}\right ){}^{\frac {1}{1-k}}\right \}\right \}\]
Maple ✓
cpu = 0.076 (sec), leaf count = 51
\[\left [y \left (x \right ) = 2^{\frac {1}{k -1}} \left (\frac {-x^{2} a k +a \,x^{2}+2 \textit {\_C1}}{x^{2}}\right )^{-\frac {1}{k -1}} x^{-\frac {2 k}{k -1}}\right ]\] Mathematica raw input
DSolve[2*y[x] + x*y'[x] == a*x^(2*k)*y[x]^k,y[x],x]
Mathematica raw output
{{y[x] -> ((a*x^(2*k))/2 - (a*k*x^(2*k))/2 + x^(-2 + 2*k)*C[1])^(1 - k)^(-1)}}
Maple raw input
dsolve(x*diff(y(x),x)+2*y(x) = a*x^(2*k)*y(x)^k, y(x))
Maple raw output
[y(x) = 2^(1/(k-1))/(((-a*k*x^2+a*x^2+2*_C1)/x^2)^(1/(k-1)))/(x^(1/(k-1)*k))^2]