ODE
\[ x y''(x)+2 y'(x)=a x^{2 k} y'(x)^k \] ODE Classification
[[_2nd_order, _missing_y]]
Book solution method
TO DO
Mathematica ✓
cpu = 56.3344 (sec), leaf count = 57
\[\left \{\left \{y(x)\to \int _1^x \left (\frac {1}{2} a K[1]^{2 k}-\frac {1}{2} a k K[1]^{2 k}+c_1 K[1]^{2 k-2}\right ){}^{\frac {1}{1-k}} \, dK[1]+c_2\right \}\right \}\]
Maple ✓
cpu = 0.32 (sec), leaf count = 47
\[ \left \{ y \left ( x \right ) =\int \!{2}^{ \left ( k-1 \right ) ^{-1}} \left ( {\frac {{x}^{-2\,k+2}}{a \left ( k-1 \right ) \left ( -{x}^{2}+{\it \_C1} \right ) }} \right ) ^{ \left ( k-1 \right ) ^{-1}}\,{\rm d}x+{\it \_C2} \right \} \] 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] -> C[2] + Integrate[((a*K[1]^(2*k))/2 - (a*k*K[1]^(2*k))/2 + C[1]*K[1]^(-2 + 2*k))^(1 - k)^(-1), {K[1], 1, x}]}}
Maple raw input
dsolve(x*diff(diff(y(x),x),x)+2*diff(y(x),x) = a*x^(2*k)*diff(y(x),x)^k, y(x),'implicit')
Maple raw output
y(x) = Int(2^(1/(k-1))*(x^(-2*k+2)/a/(k-1)/(-x^2+_C1))^(1/(k-1)),x)+_C2