ODE
\[ x^k y(x) \left (a+b x^k\right )+x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.0622507 (sec), leaf count = 147
\[\left \{\left \{y(x)\to 2^{\frac {k+1}{2 k}} x^{\frac {1}{2}-\frac {k}{2}} \left (x^k\right )^{\frac {k+1}{2 k}} e^{\frac {i \sqrt {b} x^k}{k}} \left (c_1 U\left (\frac {-\frac {i a}{\sqrt {b}}+k+1}{2 k},1+\frac {1}{k},-\frac {2 i \sqrt {b} x^k}{k}\right )+c_2 L_{-\frac {-\frac {i a}{\sqrt {b}}+k+1}{2 k}}^{\frac {1}{k}}\left (-\frac {2 i \sqrt {b} x^k}{k}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.203 (sec), leaf count = 73
\[ \left \{ y \left ( x \right ) ={x}^{{\frac {1}{2}}-{\frac {k}{2}}} \left ( {{\sl W}_{{\frac {-{\frac {i}{2}}a}{k}{\frac {1}{\sqrt {b}}}},\,{\frac {1}{2\,k}}}\left ({\frac {2\,i{x}^{k}}{k}\sqrt {b}}\right )}{\it \_C2}+{{\sl M}_{{\frac {-{\frac {i}{2}}a}{k}{\frac {1}{\sqrt {b}}}},\,{\frac {1}{2\,k}}}\left ({\frac {2\,i{x}^{k}}{k}\sqrt {b}}\right )}{\it \_C1} \right ) \right \} \] Mathematica raw input
DSolve[x^k*(a + b*x^k)*y[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> 2^((1 + k)/(2*k))*E^((I*Sqrt[b]*x^k)/k)*x^(1/2 - k/2)*(x^k)^((1 + k)/(2*k))*(C[1]*HypergeometricU[(1 - (I*a)/Sqrt[b] + k)/(2*k), 1 + k^(-1), ((-2*I)*Sqrt[b]*x^k)/k] + C[2]*LaguerreL[-(1 - (I*a)/Sqrt[b] + k)/(2*k), k^(-1), ((-2*I)*Sqrt[b]*x^k)/k])}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+x^k*(a+b*x^k)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = x^(1/2-1/2*k)*(WhittakerW(-1/2*I/k/b^(1/2)*a,1/2/k,2*I/k*b^(1/2)*x^k)*_C2+WhittakerM(-1/2*I/k/b^(1/2)*a,1/2/k,2*I/k*b^(1/2)*x^k)*_C1)