ODE
\[ -b y(x) \left (a+b x^2\right )+a y'(x)+x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.333655 (sec), leaf count = 38
\[\left \{\left \{y(x)\to e^{b x} \left (c_2 \int _1^xe^{\frac {a}{K[1]}-2 b K[1]}dK[1]+c_1\right )\right \}\right \}\]
Maple ✓
cpu = 1.595 (sec), leaf count = 180
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{\frac {-b \,x^{2}+a}{x}} \mathit {HD}\left (4 \sqrt {2}\, \sqrt {a b}, -1-4 \sqrt {2}\, \sqrt {a b}, 8 \sqrt {2}\, \sqrt {a b}, -4 \sqrt {2}\, \sqrt {a b}+1, \frac {\sqrt {2}\, \sqrt {a b}\, x -a}{\sqrt {2}\, \sqrt {a b}\, x +a}\right ) \sqrt {x}+\textit {\_C2} \,{\mathrm e}^{b x} \mathit {HD}\left (-4 \sqrt {2}\, \sqrt {a b}, -1-4 \sqrt {2}\, \sqrt {a b}, 8 \sqrt {2}\, \sqrt {a b}, -4 \sqrt {2}\, \sqrt {a b}+1, \frac {\sqrt {2}\, \sqrt {a b}\, x -a}{\sqrt {2}\, \sqrt {a b}\, x +a}\right ) \sqrt {x}\right ]\] Mathematica raw input
DSolve[-(b*(a + b*x^2)*y[x]) + a*y'[x] + x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> E^(b*x)*(C[1] + C[2]*Inactive[Integrate][E^(a/K[1] - 2*b*K[1]), {K[1], 1, x}])}}
Maple raw input
dsolve(x^2*diff(diff(y(x),x),x)+a*diff(y(x),x)-b*(b*x^2+a)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*exp((-b*x^2+a)/x)*HeunD(4*2^(1/2)*(a*b)^(1/2),-1-4*2^(1/2)*(a*b)^(1/2),8*2^(1/2)*(a*b)^(1/2),-4*2^(1/2)*(a*b)^(1/2)+1,(2^(1/2)*(a*b)^(1/2)*x-a)/(2^(1/2)*(a*b)^(1/2)*x+a))*x^(1/2)+_C2*exp(b*x)*HeunD(-4*2^(1/2)*(a*b)^(1/2),-1-4*2^(1/2)*(a*b)^(1/2),8*2^(1/2)*(a*b)^(1/2),-4*2^(1/2)*(a*b)^(1/2)+1,(2^(1/2)*(a*b)^(1/2)*x-a)/(2^(1/2)*(a*b)^(1/2)*x+a))*x^(1/2)]