ODE
\[ a x y'(x)+y(x) \left (\text {a0}+\text {b0} x+\text {c0} x^2\right )+\left (1-x^2\right ) y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.399387 (sec), leaf count = 185
\[\left \{\left \{y(x)\to \frac {1}{2} (x-1)^{-a/4} e^{\sqrt {\text {c0}} x} \left (c_2 \left (x^2-1\right )^{a/4} (x+1)^{\frac {a}{4}+1} \text {HeunC}\left [\frac {a^2}{4}+a \left (\sqrt {\text {c0}}+\frac {1}{2}\right )+\text {a0}-\text {b0}+\text {c0}+4 \sqrt {\text {c0}},4 \sqrt {\text {c0}}-2 \text {b0},\frac {a+4}{2},-\frac {a}{2},4 \sqrt {\text {c0}},\frac {x+1}{2}\right ]+2 c_1 (x-1)^{a/4} \text {HeunC}\left [a \left (-\sqrt {\text {c0}}\right )+\text {a0}-\text {b0}+\text {c0},-2 \left (a \sqrt {\text {c0}}+\text {b0}\right ),-\frac {a}{2},-\frac {a}{2},4 \sqrt {\text {c0}},\frac {x+1}{2}\right ]\right )\right \}\right \}\]
Maple ✓
cpu = 1.505 (sec), leaf count = 148
\[\left [y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{\sqrt {\mathit {c0}}\, x} \left (x^{2}-1\right )^{\frac {a}{4}} \left (\left (x -1\right ) \left (x +1\right )\right )^{\frac {a}{4}+1} \HeunC \left (4 \sqrt {\mathit {c0}}, \frac {a}{2}+1, \frac {a}{2}+1, -2 \mathit {b0} , -\mathit {a0} +\mathit {b0} -\frac {a^{2}}{8}-\mathit {c0} +\frac {1}{2}, \frac {1}{2}+\frac {x}{2}\right )+\textit {\_C2} \,{\mathrm e}^{\sqrt {\mathit {c0}}\, x} \left (\frac {1}{2}+\frac {x}{2}\right )^{-\frac {a}{4}} \left (-\frac {1}{2}+\frac {x}{2}\right )^{\frac {a}{4}+1} \left (x^{2}-1\right )^{\frac {a}{4}} \HeunC \left (4 \sqrt {\mathit {c0}}, -\frac {a}{2}-1, \frac {a}{2}+1, -2 \mathit {b0} , -\mathit {a0} +\mathit {b0} -\frac {a^{2}}{8}-\mathit {c0} +\frac {1}{2}, \frac {1}{2}+\frac {x}{2}\right )\right ]\] Mathematica raw input
DSolve[(a0 + b0*x + c0*x^2)*y[x] + a*x*y'[x] + (1 - x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (E^(Sqrt[c0]*x)*((1 + x)^(1 + a/4)*(-1 + x^2)^(a/4)*C[2]*HeunC[a^2/4 + a0 - b0 + a*(1/2 + Sqrt[c0]) + 4*Sqrt[c0] + c0, -2*b0 + 4*Sqrt[c0], (4 + a)/2, -1/2*a, 4*Sqrt[c0], (1 + x)/2] + 2*(-1 + x)^(a/4)*C[1]*HeunC[a0 - b0 - a*Sqrt[c0] + c0, -2*(b0 + a*Sqrt[c0]), -1/2*a, -1/2*a, 4*Sqrt[c0], (1 + x)/2]))/(2*(-1 + x)^(a/4))}}
Maple raw input
dsolve((-x^2+1)*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+(c0*x^2+b0*x+a0)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*exp(c0^(1/2)*x)*(x^2-1)^(1/4*a)*((x-1)*(x+1))^(1/4*a+1)*HeunC(4*c0^(1/2),1/2*a+1,1/2*a+1,-2*b0,-a0+b0-1/8*a^2-c0+1/2,1/2+1/2*x)+_C2*exp(c0^(1/2)*x)*(1/2+1/2*x)^(-1/4*a)*(-1/2+1/2*x)^(1/4*a+1)*(x^2-1)^(1/4*a)*HeunC(4*c0^(1/2),-1/2*a-1,1/2*a+1,-2*b0,-a0+b0-1/8*a^2-c0+1/2,1/2+1/2*x)]