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

ODE
\[ a (k+1) x^{k-1} y(x)+a x^k y'(x)+y''(x)=0 \] 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

\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\rm e}^{-{\frac {a{x}^{k+1}}{k+1}}}}x+{\it \_C2}\, \left ( \left ( k+1 \right ) \left ( {x}^{-{\frac {k}{2}}}a-{x}^{-{\frac {3\,k}{2}}-1}k \right ) {{\sl M}_{{\frac {-2-k}{2\,k+2}},\,{\frac {1+2\,k}{2\,k+2}}}\left (-{\frac {a{x}^{k+1}}{k+1}}\right )}-{{\sl M}_{{\frac {k}{2\,k+2}},\,{\frac {1+2\,k}{2\,k+2}}}\left (-{\frac {a{x}^{k+1}}{k+1}}\right )}{x}^{-{\frac {3\,k}{2}}-1}{k}^{2} \right ) {{\rm e}^{-{\frac {a{x}^{k+1}}{2\,k+2}}}} \right \} \] 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))