ODE
\[ a k x^{k-1} y(x)+a x^k y'(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0837096 (sec), leaf count = 74
\[\left \{\left \{y(x)\to e^{-\frac {a x^{k+1}}{k+1}} \left (c_2-\frac {c_1 x \left (-\frac {a x^{k+1}}{k+1}\right )^{-\frac {1}{k+1}} \Gamma \left (\frac {1}{k+1},-\frac {a x^{k+1}}{k+1}\right )}{k+1}\right )\right \}\right \}\]
Maple ✓
cpu = 0.054 (sec), leaf count = 216
\[ \left \{ y \left ( x \right ) ={1 \left ( - \left ( -{\frac {a{x}^{k+1}}{k+1}} \right ) ^{{\frac {-2-k}{2\,k+2}}}{\it \_C1}\,{x}^{-k}{{\rm e}^{{\frac {a{x}^{k+1}}{2\,k+2}}}} \left ( k+2 \right ) ^{2}{{\sl M}_{{\frac {k+2}{2\,k+2}},\,{\frac {2\,k+3}{2\,k+2}}}\left (-{\frac {a{x}^{k+1}}{k+1}}\right )}+ \left ( -{\frac {a{x}^{k+1}}{k+1}} \right ) ^{{\frac {-2-k}{2\,k+2}}}{\it \_C1}\,{x}^{-k}{{\rm e}^{{\frac {a{x}^{k+1}}{2\,k+2}}}} \left ( k+1 \right ) \left ( a{x}^{k+1}-k-2 \right ) {{\sl M}_{-{\frac {k}{2\,k+2}},\,{\frac {2\,k+3}{2\,k+2}}}\left (-{\frac {a{x}^{k+1}}{k+1}}\right )}+{\it \_C2} \right ) \left ( {{\rm e}^{{\frac {a{x}^{k+1}}{k+1}}}} \right ) ^{-1}} \right \} \] Mathematica raw input
DSolve[a*k*x^(-1 + k)*y[x] + a*x^k*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[2] - (x*C[1]*Gamma[(1 + k)^(-1), -((a*x^(1 + k))/(1 + k))])/((1 + k)*(-((a*x^(1 + k))/(1 + k)))^(1 + k)^(-1)))/E^((a*x^(1 + k))/(1 + k))}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+a*x^k*diff(y(x),x)+a*k*x^(k-1)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = (-(-x^(k+1)/(k+1)*a)^((-2-k)/(2*k+2))*_C1*x^(-k)*exp(a*x^(k+1)/(2*k+2))*(k+2)^2*WhittakerM((k+2)/(2*k+2),(2*k+3)/(2*k+2),-x^(k+1)/(k+1)*a)+(-x^(k+1)/(k+1)*a)^((-2-k)/(2*k+2))*_C1*x^(-k)*exp(a*x^(k+1)/(2*k+2))*(k+1)*(a*x^(k+1)-k-2)*WhittakerM(-k/(2*k+2),(2*k+3)/(2*k+2),-x^(k+1)/(k+1)*a)+_C2)/exp(x^(k+1)/(k+1)*a)