ODE
\[ a k x^{k-1} y(x)+2 a x^k y'(x)+2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0348926 (sec), leaf count = 120
\[\left \{\left \{y(x)\to c_2 \left (\frac {1}{k}+1\right )^{-\frac {1}{k+1}} k^{-\frac {1}{k+1}} a^{\frac {1}{k+1}} \left (x^k\right )^{\frac {1}{k}} \, _1F_1\left (\frac {k+2}{2 k+2};\frac {k+2}{k+1};-\frac {a \left (x^k\right )^{1+\frac {1}{k}}}{k+1}\right )+c_1 \, _1F_1\left (\frac {k}{2 k+2};\frac {k}{k+1};-\frac {a \left (x^k\right )^{1+\frac {1}{k}}}{k+1}\right )\right \}\right \}\]
Maple ✓
cpu = 0.635 (sec), leaf count = 85
\[ \left \{ y \left ( x \right ) =\sqrt {x}{{\rm e}^{-{\frac {a{x}^{k+1}}{2\,k+2}}}} \left ( {{\sl Y}_{ \left ( 2\,k+2 \right ) ^{-1}}\left ({\frac {{x}^{k+1}}{2\,k+2}\sqrt {-{a}^{2}}}\right )}{\it \_C2}+{{\sl J}_{ \left ( 2\,k+2 \right ) ^{-1}}\left ({\frac {{x}^{k+1}}{2\,k+2}\sqrt {-{a}^{2}}}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[a*k*x^(-1 + k)*y[x] + 2*a*x^k*y'[x] + 2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Hypergeometric1F1[k/(2 + 2*k), k/(1 + k), -((a*(x^k)^(1 + k^(-1)))/(1 + k))] + (a^(1 + k)^(-1)*(x^k)^k^(-1)*C[2]*Hypergeometric1F1[(2 + k)/(2 + 2*k), (2 + k)/(1 + k), -((a*(x^k)^(1 + k^(-1)))/(1 + k))])/((1 + k^(-1))^(1 + k)^(-1)*k^(1 + k)^(-1))}}
Maple raw input
dsolve(2*diff(diff(y(x),x),x)+2*a*x^k*diff(y(x),x)+a*k*x^(k-1)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = x^(1/2)*exp(-a*x^(k+1)/(2*k+2))*(BesselY(1/(2*k+2),(-a^2)^(1/2)*x^(k+1)/(2*k+2))*_C2+BesselJ(1/(2*k+2),(-a^2)^(1/2)*x^(k+1)/(2*k+2))*_C1)