4.27.6 $a(k+1)x^{k-1}y(x)+a{x}^k{y}^{\prime }(x)+y^{″}(x)=0$

ODE
$$a(k+1)x^{k-1}y(x)+a{x}^k{y}^{\prime }(x)+y^{″}(x)=0$$ ODE Classification

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✗
cpu = 1.40809 (sec), leaf count = 0 , could not solve

DSolve[a*(1 + k)*x^(-1 + k)*y[x] + a*x^k*Derivative[1][y][x] + Derivative[2][y][x] == 0, y[x], x]

Maple ✓
cpu = 0.39 (sec), leaf count = 151

$$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}{\mathrm{e}}^{-\frac{a{x}^{k+1}}{k+1}}x+\mathit{\_}\mathit{C}\mathit{2}((k+1)(x^{-\frac{k}{2}}a-x^{-\frac{3k}{2}-1}k){\mathit{M}}_{\frac{-2-k}{2k+2},\frac{1+2k}{2k+2}}(-\frac{a{x}^{k+1}}{k+1})-{\mathit{M}}_{\frac{k}{2k+2},\frac{1+2k}{2k+2}}(-\frac{a{x}^{k+1}}{k+1})x^{-\frac{3k}{2}-1}{k}^2){\mathrm{e}}^{-\frac{a{x}^{k+1}}{2k+2}}\}$$ Mathematica raw input

DSolve[a*(1 + k)*x^(-1 + k)*y[x] + a*x^k*y'[x] + y''[x] == 0,y[x],x]

Mathematica raw output

DSolve[a*(1 + k)*x^(-1 + k)*y[x] + a*x^k*Derivative[1][y][x] + Derivative[2][y][
x] == 0, y[x], x]

Maple raw input

dsolve(diff(diff(y(x),x),x)+a*x^k*diff(y(x),x)+a*(k+1)*x^(k-1)*y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = _C1*exp(-x^(k+1)/(k+1)*a)*x+_C2*((k+1)*(x^(-1/2*k)*a-x^(-3/2*k-1)*k)*Whit
takerM((-2-k)/(2*k+2),(1+2*k)/(2*k+2),-x^(k+1)/(k+1)*a)-WhittakerM(k/(2*k+2),(1+
2*k)/(2*k+2),-x^(k+1)/(k+1)*a)*x^(-3/2*k-1)*k^2)*exp(-a*x^(k+1)/(2*k+2))