ODE
\[ y''(x)=a \left (x y'(x)-y(x)\right )^k \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 58.9088 (sec), leaf count = 59
\[\left \{\left \{y(x)\to x \left (\int _1^x \left (\frac {1}{2} a K[2]^{2 k}-\frac {1}{2} a k K[2]^{2 k}+c_1 K[2]^{2 k-2}\right ){}^{\frac {1}{1-k}} \, dK[2]+c_2\right )\right \}\right \}\]
Maple ✓
cpu = 1.187 (sec), leaf count = 69
\[ \left \{ {\frac {1}{x} \left ( -{\it \_C2}\,x-\int \!-{\frac {{x}^{2} \left ( k-1 \right ) a-{\it \_C1}}{2\,{x}^{2}} \left ( - \left ( {x}^{2} \left ( k-1 \right ) a-{\it \_C1} \right ) ^{-1} \right ) ^{{\frac {k}{k-1}}}{2}^{{\frac {k}{k-1}}}}\,{\rm d}xx+y \left ( x \right ) \right ) }=0 \right \} \] Mathematica raw input
DSolve[y''[x] == a*(-y[x] + x*y'[x])^k,y[x],x]
Mathematica raw output
{{y[x] -> x*(C[2] + Integrate[((a*K[2]^(2*k))/2 - (a*k*K[2]^(2*k))/2 + C[1]*K[2]^(-2 + 2*k))^(1 - k)^(-1), {K[2], 1, x}])}}
Maple raw input
dsolve(diff(diff(y(x),x),x) = a*(x*diff(y(x),x)-y(x))^k, y(x),'implicit')
Maple raw output
(-_C2*x-Int(-1/2*(-1/(x^2*(k-1)*a-_C1))^(k/(k-1))*(x^2*(k-1)*a-_C1)*2^(k/(k-1))/x^2,x)*x+y(x))/x = 0