ODE
\[ a k x^{k-1} y(x)+a x^k y'(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.170391 (sec), leaf count = 74
\[\left \{\left \{y(x)\to e^{-\frac {a x^{k+1}}{k+1}} \left (c_2-\frac {c_1 x \left (-\frac {a x^{k+1}}{k+1}\right )^{-\frac {1}{k+1}} \Gamma \left (\frac {1}{k+1},-\frac {a x^{k+1}}{k+1}\right )}{k+1}\right )\right \}\right \}\]
Maple ✓
cpu = 0.166 (sec), leaf count = 482
\[\left [y \left (x \right ) = \left (x^{k +1} \WhittakerM \left (\frac {1}{k +1}-\frac {k +2}{2 \left (k +1\right )}, \frac {k +2}{2 k +2}+\frac {1}{2}, \frac {x^{k +1} a}{-k -1}\right ) a k +x^{k +1} \WhittakerM \left (\frac {1}{k +1}-\frac {k +2}{2 \left (k +1\right )}, \frac {k +2}{2 k +2}+\frac {1}{2}, \frac {x^{k +1} a}{-k -1}\right ) a -\WhittakerM \left (\frac {1}{k +1}-\frac {k +2}{2 \left (k +1\right )}, \frac {k +2}{2 k +2}+\frac {1}{2}, \frac {x^{k +1} a}{-k -1}\right ) k^{2}-\WhittakerM \left (\frac {1}{k +1}-\frac {k +2}{2 \left (k +1\right )}+1, \frac {k +2}{2 k +2}+\frac {1}{2}, \frac {x^{k +1} a}{-k -1}\right ) k^{2}-3 \WhittakerM \left (\frac {1}{k +1}-\frac {k +2}{2 \left (k +1\right )}, \frac {k +2}{2 k +2}+\frac {1}{2}, \frac {x^{k +1} a}{-k -1}\right ) k -4 \WhittakerM \left (\frac {1}{k +1}-\frac {k +2}{2 \left (k +1\right )}+1, \frac {k +2}{2 k +2}+\frac {1}{2}, \frac {x^{k +1} a}{-k -1}\right ) k -2 \WhittakerM \left (\frac {1}{k +1}-\frac {k +2}{2 \left (k +1\right )}, \frac {k +2}{2 k +2}+\frac {1}{2}, \frac {x^{k +1} a}{-k -1}\right )-4 \WhittakerM \left (\frac {1}{k +1}-\frac {k +2}{2 \left (k +1\right )}+1, \frac {k +2}{2 k +2}+\frac {1}{2}, \frac {x^{k +1} a}{-k -1}\right )\right ) {\mathrm e}^{-\frac {x^{k +1} a}{2 \left (-k -1\right )}} \left (\frac {x^{k +1} a}{-k -1}\right )^{-\frac {k +2}{2 \left (k +1\right )}} x^{\frac {1}{k +1}+\frac {k}{k +1}-k -1} {\mathrm e}^{-\frac {x^{k +1} a}{k +1}} \textit {\_C1} +\textit {\_C2} \,{\mathrm e}^{-\frac {x^{k +1} a}{k +1}}\right ]\] 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))
Maple raw output
[y(x) = (x^(k+1)*WhittakerM(1/(k+1)-1/2*(k+2)/(k+1),1/2*(k+2)/(k+1)+1/2,x^(k+1)/(-k-1)*a)*a*k+x^(k+1)*WhittakerM(1/(k+1)-1/2*(k+2)/(k+1),1/2*(k+2)/(k+1)+1/2,x^(k+1)/(-k-1)*a)*a-WhittakerM(1/(k+1)-1/2*(k+2)/(k+1),1/2*(k+2)/(k+1)+1/2,x^(k+1)/(-k-1)*a)*k^2-WhittakerM(1/(k+1)-1/2*(k+2)/(k+1)+1,1/2*(k+2)/(k+1)+1/2,x^(k+1)/(-k-1)*a)*k^2-3*WhittakerM(1/(k+1)-1/2*(k+2)/(k+1),1/2*(k+2)/(k+1)+1/2,x^(k+1)/(-k-1)*a)*k-4*WhittakerM(1/(k+1)-1/2*(k+2)/(k+1)+1,1/2*(k+2)/(k+1)+1/2,x^(k+1)/(-k-1)*a)*k-2*WhittakerM(1/(k+1)-1/2*(k+2)/(k+1),1/2*(k+2)/(k+1)+1/2,x^(k+1)/(-k-1)*a)-4*WhittakerM(1/(k+1)-1/2*(k+2)/(k+1)+1,1/2*(k+2)/(k+1)+1/2,x^(k+1)/(-k-1)*a))*exp(-1/2*x^(k+1)/(-k-1)*a)*(x^(k+1)/(-k-1)*a)^(-1/2*(k+2)/(k+1))*x^(1/(k+1)+1/(k+1)*k-k-1)/exp(x^(k+1)/(k+1)*a)*_C1+_C2/exp(x^(k+1)/(k+1)*a)]