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 = 0.713559 (sec), leaf count = 91
\[\left \{\left \{y(x)\to x \left (c_2-x \left (2-\frac {a (k-1) x^2}{c_1}\right ){}^{\frac {1}{k-1}} \left (x^{2 k-2} \left (-a (k-1) x^2+2 c_1\right )\right ){}^{\frac {1}{1-k}} \, _2F_1\left (-\frac {1}{2},\frac {1}{k-1};\frac {1}{2};\frac {a (k-1) x^2}{2 c_1}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 2.079 (sec), leaf count = 123
\[\left [y \left (x \right ) = \left (\int \left (-\frac {2^{\frac {k}{k -1}} \left (\frac {1}{-a k \,x^{2}+a \,x^{2}+\textit {\_C1}}\right )^{\frac {k}{k -1}} a k}{2}+\frac {2^{\frac {k}{k -1}} \left (\frac {1}{-a k \,x^{2}+a \,x^{2}+\textit {\_C1}}\right )^{\frac {k}{k -1}} a}{2}+\frac {2^{\frac {k}{k -1}} \left (\frac {1}{-a k \,x^{2}+a \,x^{2}+\textit {\_C1}}\right )^{\frac {k}{k -1}} \textit {\_C1}}{2 x^{2}}\right )d x +\textit {\_C2} \right ) x\right ]\] Mathematica raw input
DSolve[y''[x] == a*(-y[x] + x*y'[x])^k,y[x],x]
Mathematica raw output
{{y[x] -> x*(C[2] - 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(diff(diff(y(x),x),x) = a*(x*diff(y(x),x)-y(x))^k, y(x))
Maple raw output
[y(x) = (Int(-1/2*2^(k/(k-1))*(1/(-a*k*x^2+a*x^2+_C1))^(k/(k-1))*a*k+1/2*2^(k/(k-1))*(1/(-a*k*x^2+a*x^2+_C1))^(k/(k-1))*a+1/2/x^2*2^(k/(k-1))*(1/(-a*k*x^2+a*x^2+_C1))^(k/(k-1))*_C1,x)+_C2)*x]