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 = 0.496006 (sec), leaf count = 89
\[\left \{\left \{y(x)\to c_2-x \left (2-\frac {a (k-1) x^2}{c_1}\right ){}^{\frac {1}{k-1}} \left (x^{2 k-2} \left (-a (k-1) x^2+2 c_1\right )\right ){}^{\frac {1}{1-k}} \, _2F_1\left (-\frac {1}{2},\frac {1}{k-1};\frac {1}{2};\frac {a (k-1) x^2}{2 c_1}\right )\right \}\right \}\]
Maple ✓
cpu = 1.046 (sec), leaf count = 46
\[\left [y \left (x \right ) = \int 2^{\frac {1}{k -1}} \left (\frac {x^{2-2 k}}{-a k \,x^{2}+a \,x^{2}+\textit {\_C1}}\right )^{\frac {1}{k -1}}d x +\textit {\_C2}\right ]\] 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] - x*(2 - (a*(-1 + k)*x^2)/C[1])^(-1 + k)^(-1)*(x^(-2 + 2*k)*(-(a*(-1 + k)*x^2) + 2*C[1]))^(1 - k)^(-1)*Hypergeometric2F1[-1/2, (-1 + k)^(-1), 1/2, (a*(-1 + k)*x^2)/(2*C[1])]}}
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))
Maple raw output
[y(x) = Int(2^(1/(k-1))*(x^(2-2*k)/(-a*k*x^2+a*x^2+_C1))^(1/(k-1)),x)+_C2]