4.27.7 $ak{x}^{k-1}y(x)+a{x}^k{y}^{\prime }(x)+y^{″}(x)=0$

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

$$\left\{\{y(x)\to e^{-\frac{a{x}^{k+1}}{k+1}}(c_2-\frac{c_{1}x{(-\frac{a{x}^{k+1}}{k+1})}^{-\frac{1}{k+1}}\mathrm{\Gamma }(\frac{1}{k+1},-\frac{a{x}^{k+1}}{k+1})}{k+1})\}\right\}$$

Maple ✓
cpu = 0.054 (sec), leaf count = 216

$$\{y\left(x\right)=1(-{(-\frac{a{x}^{k+1}}{k+1})}^{\frac{-2-k}{2k+2}}\mathit{\_}\mathit{C}\mathit{1}{x}^{-k}{\mathrm{e}}^{\frac{a{x}^{k+1}}{2k+2}}{(k+2)}^2{\mathit{M}}_{\frac{k+2}{2k+2},\frac{2k+3}{2k+2}}(-\frac{a{x}^{k+1}}{k+1})+{(-\frac{a{x}^{k+1}}{k+1})}^{\frac{-2-k}{2k+2}}\mathit{\_}\mathit{C}\mathit{1}{x}^{-k}{\mathrm{e}}^{\frac{a{x}^{k+1}}{2k+2}}(k+1)(a{x}^{k+1}-k-2){\mathit{M}}_{-\frac{k}{2k+2},\frac{2k+3}{2k+2}}(-\frac{a{x}^{k+1}}{k+1})+\mathit{\_}\mathit{C}\mathit{2}){\left({\mathrm{e}}^{\frac{a{x}^{k+1}}{k+1}}\right)}^{-1}\}$$ 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)*Wh
ittakerM(-k/(2*k+2),(2*k+3)/(2*k+2),-x^(k+1)/(k+1)*a)+_C2)/exp(x^(k+1)/(k+1)*a)