ODE
\[ x^2 y''(x)-y(x) \left ((a-1) a-b x^k\right )=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.191135 (sec), leaf count = 116
\[\left \{\left \{y(x)\to k^{-1/k} b^{\left .\frac {1}{2}\right /k} \left (x^k\right )^{\left .\frac {1}{2}\right /k} \left (c_1 \Gamma \left (\frac {-2 a+k+1}{k}\right ) J_{\frac {1-2 a}{k}}\left (\frac {2 \sqrt {b} \sqrt {x^k}}{k}\right )+c_2 \Gamma \left (\frac {2 a+k-1}{k}\right ) J_{\frac {2 a-1}{k}}\left (\frac {2 \sqrt {b} \sqrt {x^k}}{k}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.274 (sec), leaf count = 69
\[\left [y \left (x \right ) = \textit {\_C1} \sqrt {x}\, \BesselJ \left (\frac {\sqrt {\left (2 a -1\right )^{2}}}{k}, \frac {2 \sqrt {b}\, x^{\frac {k}{2}}}{k}\right )+\textit {\_C2} \sqrt {x}\, \BesselY \left (\frac {\sqrt {\left (2 a -1\right )^{2}}}{k}, \frac {2 \sqrt {b}\, x^{\frac {k}{2}}}{k}\right )\right ]\] Mathematica raw input
DSolve[-(((-1 + a)*a - b*x^k)*y[x]) + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (b^(1/(2*k))*(x^k)^(1/(2*k))*(BesselJ[(1 - 2*a)/k, (2*Sqrt[b]*Sqrt[x^k])/k]*C[1]*Gamma[(1 - 2*a + k)/k] + BesselJ[(-1 + 2*a)/k, (2*Sqrt[b]*Sqrt[x^k])/k]*C[2]*Gamma[(-1 + 2*a + k)/k]))/k^k^(-1)}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)-(a*(a-1)-b*x^k)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*x^(1/2)*BesselJ(((2*a-1)^2)^(1/2)/k,2*b^(1/2)/k*x^(1/2*k))+_C2*x^(1/2)*BesselY(((2*a-1)^2)^(1/2)/k,2*b^(1/2)/k*x^(1/2*k))]