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 = 0.0398959 (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 = 0.047 (sec), leaf count = 67
\[ \left \{ y \left ( x \right ) =\sqrt {x} \left ( {{\sl Y}_{{\frac {1}{k}\sqrt { \left ( 2\,a-1 \right ) ^{2}}}}\left (2\,{\frac {\sqrt {b}{x}^{k/2}}{k}}\right )}{\it \_C2}+{{\sl J}_{{\frac {1}{k}\sqrt { \left ( 2\,a-1 \right ) ^{2}}}}\left (2\,{\frac {\sqrt {b}{x}^{k/2}}{k}}\right )}{\it \_C1} \right ) \right \} \] 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),'implicit')
Maple raw output
y(x) = x^(1/2)*(BesselY(((2*a-1)^2)^(1/2)/k,2*b^(1/2)/k*x^(1/2*k))*_C2+BesselJ(((2*a-1)^2)^(1/2)/k,2*b^(1/2)/k*x^(1/2*k))*_C1)