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 = 12.1458 (sec), leaf count = 37
\[\left \{\left \{y(x)\to e^{b x} \left (c_2 \int _1^x e^{\frac {a}{K[1]}-2 b K[1]} \, dK[1]+c_1\right )\right \}\right \}\]
Maple ✓
cpu = 0.334 (sec), leaf count = 178
\[ \left \{ y \left ( x \right ) =\sqrt {x} \left ( {\it HeunD} \left ( -4\,\sqrt {2}\sqrt {ab},-1-4\,\sqrt {2}\sqrt {ab},8\,\sqrt {2}\sqrt {ab},-4\,\sqrt {2}\sqrt {ab}+1,{1 \left ( \sqrt {2}\sqrt {ab}x-a \right ) \left ( \sqrt {2}\sqrt {ab}x+a \right ) ^{-1}} \right ) {{\rm e}^{bx}}{\it \_C2}+{\it HeunD} \left ( 4\,\sqrt {2}\sqrt {ab},-1-4\,\sqrt {2}\sqrt {ab},8\,\sqrt {2}\sqrt {ab},-4\,\sqrt {2}\sqrt {ab}+1,{1 \left ( \sqrt {2}\sqrt {ab}x-a \right ) \left ( \sqrt {2}\sqrt {ab}x+a \right ) ^{-1}} \right ) {{\rm e}^{{\frac {-b{x}^{2}+a}{x}}}}{\it \_C1} \right ) \right \} \] 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]*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),'implicit')
Maple raw output
y(x) = x^(1/2)*(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))*exp(b*x)*_C2+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))*exp((-b*x^2+a)/x)*_C1)