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 = 2.53225 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\left (-\text {c0} \unicode {f817}^2-\text {b0} \unicode {f817}-\text {a0}\right ) \unicode {f818}(\unicode {f817})-\unicode {f817} a \unicode {f818}'(\unicode {f817})+\left (\unicode {f817}^2-1\right ) \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\]
Maple ✓
cpu = 0.254 (sec), leaf count = 134
\[ \left \{ y \left ( x \right ) = \left ( {x}^{2}-1 \right ) ^{{\frac {a}{4}}}{{\rm e}^{\sqrt {{\it c0}}x}} \left ( \left ( {\frac {1}{2}}+{\frac {x}{2}} \right ) ^{-{\frac {a}{4}}}{\it HeunC} \left ( 4\,\sqrt {{\it c0}},-{\frac {a}{2}}-1,{\frac {a}{2}}+1,-2\,{\it b0},-{\it a0}+{\it b0}-{\frac {{a}^{2}}{8}}-{\it c0}+{\frac {1}{2}},{\frac {1}{2}}+{\frac {x}{2}} \right ) \left ( -{\frac {1}{2}}+{\frac {x}{2}} \right ) ^{{\frac {a}{4}}+1}{\it \_C2}+ \left ( \left ( -1+x \right ) \left ( 1+x \right ) \right ) ^{{\frac {a}{4}}+1}{\it HeunC} \left ( 4\,\sqrt {{\it c0}},{\frac {a}{2}}+1,{\frac {a}{2}}+1,-2\,{\it b0},-{\it a0}+{\it b0}-{\frac {{a}^{2}}{8}}-{\it c0}+{\frac {1}{2}},{\frac {1}{2}}+{\frac {x}{2}} \right ) {\it \_C1} \right ) \right \} \] 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] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {(-a0 - \[FormalX]*b0 - \[FormalX]^2*c0)*\[FormalY][\[FormalX]] - \[FormalX]*a*Derivative[1][\[FormalY]][\[FormalX]] + (-1 + \[FormalX]^2)*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][0] == C[1], Derivative[1][\[FormalY]][0] == C[2]}]][x]}}
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),'implicit')
Maple raw output
y(x) = (x^2-1)^(1/4*a)*exp(c0^(1/2)*x)*((1/2+1/2*x)^(-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)*(-1/2+1/2*x)^(1/4*a+1)*_C2+((-1+x)*(1+x))^(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)*_C1)