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 = 0.206505 (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 = 1.768 (sec), leaf count = 79
\[\left [y \left (x \right ) = \textit {\_C1} \,x^{\frac {1}{2}-\frac {k}{2}} \WhittakerM \left (-\frac {i a}{2 \sqrt {b}\, k}, \frac {1}{2 k}, \frac {2 i \sqrt {b}\, x^{k}}{k}\right )+\textit {\_C2} \,x^{\frac {1}{2}-\frac {k}{2}} \WhittakerW \left (-\frac {i a}{2 \sqrt {b}\, k}, \frac {1}{2 k}, \frac {2 i \sqrt {b}\, x^{k}}{k}\right )\right ]\] 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[-1/2*(1 - (I*a)/Sqrt[b] + k)/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))
Maple raw output
[y(x) = _C1*x^(1/2-1/2*k)*WhittakerM(-1/2*I/b^(1/2)/k*a,1/2/k,2*I*b^(1/2)/k*x^k)+_C2*x^(1/2-1/2*k)*WhittakerW(-1/2*I/b^(1/2)/k*a,1/2/k,2*I*b^(1/2)/k*x^k)]