ODE
\[ y''(x) \left (\text {a0}+\text {b0} x+\text {c0} x^2\right )+(\text {a1}+\text {b1} x) y'(x)+2 \text {a2} y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✗
cpu = 16.2114 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{2 \text {a2} \unicode {f818}(\unicode {f817})+(\text {a1}+\unicode {f817} \text {b1}) \unicode {f818}'(\unicode {f817})+\left (\text {c0} \unicode {f817}^2+\text {b0} \unicode {f817}+\text {a0}\right ) \unicode {f818}''(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\]
Maple ✓
cpu = 0.245 (sec), leaf count = 507
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}({\frac {1}{2\,{\it c0}} \left ( -{\it c0}+{\it b1}+\sqrt {{{\it c0}}^{2}+ \left ( -8\,{\it a2}-2\,{\it b1} \right ) {\it c0}+{{\it b1}}^{2}} \right ) },-{\frac {1}{2\,{\it c0}} \left ( {\it c0}-{\it b1}+\sqrt {{{\it c0}}^{2}+ \left ( -8\,{\it a2}-2\,{\it b1} \right ) {\it c0}+{{\it b1}}^{2}} \right ) };\,{\frac {1}{2\,{{\it c0}}^{2}} \left ( {\it b1}\,\sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}{\it c0}-2\,{\it c0}\,{\it a1}+{\it b1}\,{\it b0} \right ) {\frac {1}{\sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}}}};\,{\frac {1}{8\,{\it a0}\,{\it c0}-2\,{{\it b0}}^{2}} \left ( -2\,\sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}x{{\it c0}}^{2}-\sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}{\it b0}\,{\it c0}+4\,{\it a0}\,{\it c0}-{{\it b0}}^{2} \right ) })}+{\it \_C2}\, \left ( 2\,\sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}x{{\it c0}}^{2}+\sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}{\it b0}\,{\it c0}-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2} \right ) ^{{\frac {1}{{{\it c0}}^{2}} \left ( -{\frac {{\it c0}\, \left ( {\it b1}-2\,{\it c0} \right ) }{2}\sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}}+{\it c0}\,{\it a1}-{\frac {{\it b1}\,{\it b0}}{2}} \right ) {\frac {1}{\sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}}}}}{\mbox {$_2$F$_1$}({\frac {1}{{{\it c0}}^{2}} \left ( {\frac {{\it c0}}{2} \left ( {\it c0}-\sqrt {{{\it c0}}^{2}+ \left ( -8\,{\it a2}-2\,{\it b1} \right ) {\it c0}+{{\it b1}}^{2}} \right ) \sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}}+{\it c0}\,{\it a1}-{\frac {{\it b1}\,{\it b0}}{2}} \right ) {\frac {1}{\sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}}}},{\frac {1}{{{\it c0}}^{2}} \left ( {\frac {{\it c0}}{2} \left ( {\it c0}+\sqrt {{{\it c0}}^{2}+ \left ( -8\,{\it a2}-2\,{\it b1} \right ) {\it c0}+{{\it b1}}^{2}} \right ) \sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}}+{\it c0}\,{\it a1}-{\frac {{\it b1}\,{\it b0}}{2}} \right ) {\frac {1}{\sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}}}};\,{\frac {1}{{{\it c0}}^{2}} \left ( -{\frac {{\it c0}\, \left ( {\it b1}-4\,{\it c0} \right ) }{2}\sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}}+{\it c0}\,{\it a1}-{\frac {{\it b1}\,{\it b0}}{2}} \right ) {\frac {1}{\sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}}}};\,{\frac {1}{8\,{\it a0}\,{\it c0}-2\,{{\it b0}}^{2}} \left ( -2\,\sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}x{{\it c0}}^{2}-\sqrt {{\frac {-4\,{\it a0}\,{\it c0}+{{\it b0}}^{2}}{{{\it c0}}^{2}}}}{\it b0}\,{\it c0}+4\,{\it a0}\,{\it c0}-{{\it b0}}^{2} \right ) })} \right \} \] Mathematica raw input
DSolve[2*a2*y[x] + (a1 + b1*x)*y'[x] + (a0 + b0*x + c0*x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> DifferentialRoot[Function[{\[FormalY], \[FormalX]}, {2*a2*\[FormalY][\[FormalX]] + (a1 + \[FormalX]*b1)*Derivative[1][\[FormalY]][\[FormalX]] + (a0 + \[FormalX]*b0 + \[FormalX]^2*c0)*Derivative[2][\[FormalY]][\[FormalX]] == 0, \[FormalY][0] == C[1], Derivative[1][\[FormalY]][0] == C[2]}]][x]}}
Maple raw input
dsolve((c0*x^2+b0*x+a0)*diff(diff(y(x),x),x)+(b1*x+a1)*diff(y(x),x)+2*a2*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*hypergeom([1/2/c0*(-c0+b1+(c0^2+(-8*a2-2*b1)*c0+b1^2)^(1/2)), -1/2/c0*(c0-b1+(c0^2+(-8*a2-2*b1)*c0+b1^2)^(1/2))],[1/2*(b1*((-4*a0*c0+b0^2)/c0^2)^(1/2)*c0-2*c0*a1+b1*b0)/c0^2/((-4*a0*c0+b0^2)/c0^2)^(1/2)],(-2*((-4*a0*c0+b0^2)/c0^2)^(1/2)*x*c0^2-((-4*a0*c0+b0^2)/c0^2)^(1/2)*b0*c0+4*a0*c0-b0^2)/(8*a0*c0-2*b0^2))+_C2*(2*((-4*a0*c0+b0^2)/c0^2)^(1/2)*x*c0^2+((-4*a0*c0+b0^2)/c0^2)^(1/2)*b0*c0-4*a0*c0+b0^2)^(1/((-4*a0*c0+b0^2)/c0^2)^(1/2)*(-1/2*c0*(b1-2*c0)*((-4*a0*c0+b0^2)/c0^2)^(1/2)+c0*a1-1/2*b1*b0)/c0^2)*hypergeom([(1/2*c0*(c0-(c0^2+(-8*a2-2*b1)*c0+b1^2)^(1/2))*((-4*a0*c0+b0^2)/c0^2)^(1/2)+c0*a1-1/2*b1*b0)/((-4*a0*c0+b0^2)/c0^2)^(1/2)/c0^2, (1/2*c0*(c0+(c0^2+(-8*a2-2*b1)*c0+b1^2)^(1/2))*((-4*a0*c0+b0^2)/c0^2)^(1/2)+c0*a1-1/2*b1*b0)/((-4*a0*c0+b0^2)/c0^2)^(1/2)/c0^2],[1/((-4*a0*c0+b0^2)/c0^2)^(1/2)*(-1/2*c0*(b1-4*c0)*((-4*a0*c0+b0^2)/c0^2)^(1/2)+c0*a1-1/2*b1*b0)/c0^2],(-2*((-4*a0*c0+b0^2)/c0^2)^(1/2)*x*c0^2-((-4*a0*c0+b0^2)/c0^2)^(1/2)*b0*c0+4*a0*c0-b0^2)/(8*a0*c0-2*b0^2))