ODE
\[ -a x^{k-1} y(x)+a x^k y'(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.176447 (sec), leaf count = 90
\[\left \{\left \{y(x)\to \frac {\left (\frac {1}{k}+1\right )^{-\frac {1}{k+1}} k^{-\frac {1}{k+1}} a^{\frac {1}{k+1}} \left (x^k\right )^{\frac {1}{k}} \left (c_2 (-1)^{\frac {1}{k+1}} (k+1)-c_1 \Gamma \left (-\frac {1}{k+1},0,\frac {a \left (x^k\right )^{1+\frac {1}{k}}}{k+1}\right )\right )}{k+1}\right \}\right \}\]
Maple ✓
cpu = 1.089 (sec), leaf count = 133
\[\left [y \left (x \right ) = \textit {\_C1} x +\textit {\_C2} \left (\left (k +1\right ) \left (x^{-\frac {k}{2}} a +k \,x^{-\frac {3 k}{2}-1}\right ) \WhittakerM \left (\frac {-k -2}{2 k +2}, \frac {1+2 k}{2 k +2}, \frac {x^{k +1} a}{k +1}\right )+x^{-\frac {3 k}{2}-1} \WhittakerM \left (\frac {k}{2 k +2}, \frac {1+2 k}{2 k +2}, \frac {x^{k +1} a}{k +1}\right ) k^{2}\right ) {\mathrm e}^{-\frac {a \,x^{k +1}}{2 k +2}}\right ]\] 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]*Gamma[-(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))
Maple raw output
[y(x) = _C1*x+_C2*((k+1)*(x^(-1/2*k)*a+k*x^(-3/2*k-1))*WhittakerM((-k-2)/(2*k+2),(1+2*k)/(2*k+2),x^(k+1)/(k+1)*a)+x^(-3/2*k-1)*WhittakerM(k/(2*k+2),(1+2*k)/(2*k+2),x^(k+1)/(k+1)*a)*k^2)*exp(-a*x^(k+1)/(2*k+2))]