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 = 0.81184 (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 = 1.301 (sec), leaf count = 151

\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-\frac {x^{k +1} a}{k +1}} x +\textit {\_C2} \left (\left (k +1\right ) \left (x^{-\frac {k}{2}} a -k \,x^{-\frac {3 k}{2}-1}\right ) \WhittakerM \left (\frac {-k -2}{2 k +2}, \frac {1+2 k}{2 k +2}, -\frac {x^{k +1} a}{k +1}\right )-\WhittakerM \left (\frac {k}{2 k +2}, \frac {1+2 k}{2 k +2}, -\frac {x^{k +1} a}{k +1}\right ) x^{-\frac {3 k}{2}-1} k^{2}\right ) {\mathrm e}^{-\frac {a \,x^{k +1}}{2 k +2}}\right ]\] 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))

Maple raw output

[y(x) = _C1*exp(-x^(k+1)/(k+1)*a)*x+_C2*((k+1)*(x^(-1/2*k)*a-k*x^(-3/2*k-1))*Whi
ttakerM((-k-2)/(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))]