4.4.44 \(x y'(x)+2 y(x)=a x^{2 k} y(x)^k\)

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 = 0.0265826 (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 = 0.025 (sec), leaf count = 34

\[ \left \{ -{\frac { \left ({x}^{k} \right ) ^{2}{\it \_C1}}{{x}^{2}}}+ \left (y \relax (x ) \right ) ^{1-k}+{\frac {{x}^{2\,k}a \left (k-1 \right ) }{2}}=0 \right \} \] 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),'implicit')

Maple raw output

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