ODE
\[ a x^k y'(x)+b x^{k-1} y(x)+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0472613 (sec), leaf count = 120
\[\left \{\left \{y(x)\to c_2 \left (\frac {1}{k}+1\right )^{-\frac {1}{k+1}} k^{-\frac {1}{k+1}} a^{\frac {1}{k+1}} \left (x^k\right )^{\frac {1}{k}} \, _1F_1\left (\frac {a+b}{k a+a};\frac {k+2}{k+1};-\frac {a \left (x^k\right )^{1+\frac {1}{k}}}{k+1}\right )+c_1 \, _1F_1\left (\frac {b}{k a+a};\frac {k}{k+1};-\frac {a \left (x^k\right )^{1+\frac {1}{k}}}{k+1}\right )\right \}\right \}\]
Maple ✓
cpu = 0.211 (sec), leaf count = 105
\[ \left \{ y \left ( x \right ) ={{\rm e}^{-{\frac {a{x}^{k+1}}{k+1}}}}x \left ( {{\sl U}\left ({\frac {a \left ( k+1 \right ) -b}{a \left ( k+1 \right ) }},\,{\frac {k+2}{k+1}},\,{\frac {a{x}^{k+1}}{k+1}}\right )}{\it \_C2}+{{\sl M}\left ({\frac {a \left ( k+1 \right ) -b}{a \left ( k+1 \right ) }},\,{\frac {k+2}{k+1}},\,{\frac {a{x}^{k+1}}{k+1}}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[b*x^(-1 + k)*y[x] + a*x^k*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Hypergeometric1F1[b/(a + a*k), k/(1 + k), -((a*(x^k)^(1 + k^(-1)))/(1 + k))] + (a^(1 + k)^(-1)*(x^k)^k^(-1)*C[2]*Hypergeometric1F1[(a + b)/(a + a*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(diff(diff(y(x),x),x)+a*x^k*diff(y(x),x)+b*x^(k-1)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = exp(-x^(k+1)/(k+1)*a)*x*(KummerU((a*(k+1)-b)/a/(k+1),(k+2)/(k+1),x^(k+1)/(k+1)*a)*_C2+KummerM((a*(k+1)-b)/a/(k+1),(k+2)/(k+1),x^(k+1)/(k+1)*a)*_C1)