4.27.8 $-a{x}^{k-1}y(x)+a{x}^k{y}^{\prime }(x)+y^{″}(x)=0$

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

[[_2nd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica ✓
cpu = 0.0477762 (sec), leaf count = 90

$$\left\{\{y(x)\to \frac{{(\frac{1}{k}+1)}^{-\frac{1}{k+1}}{k}^{-\frac{1}{k+1}}{a}^{\frac{1}{k+1}}{\left(x^k\right)}^{\frac{1}{k}}(c_2(-1{)}^{\frac{1}{k+1}}(k+1)-c_1\mathrm{\Gamma }(-\frac{1}{k+1},0,\frac{a{\left(x^k\right)}^{1+\frac{1}{k}}}{k+1}))}{k+1}\}\right\}$$

Maple ✓
cpu = 0.248 (sec), leaf count = 133

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

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

Mathematica raw output

{{y[x] -> (a^(1 + k)^(-1)*(x^k)^k^(-1)*((-1)^(1 + k)^(-1)*(1 + k)*C[2] - C[1]*Ga
mma[-(1 + k)^(-1), 0, (a*(x^k)^(1 + k^(-1)))/(1 + k)]))/((1 + k^(-1))^(1 + k)^(-
1)*k^(1 + k)^(-1)*(1 + k))}}

Maple raw input

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

Maple raw output

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