ODE
\[ (\text {a1}-x) (\text {a2}-x) (\text {a3}-x) y''(x)+y'(x) \left (\text {b0}+\text {b1} x+\text {b2} x^2\right )+y(x) (\text {c0}+\text {c1} x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 3.95979 (sec), leaf count = 1131
\[\left \{\left \{y(x)\to c_2 \text {HeunG}\left [\frac {\text {a1}-\text {a3}}{\text {a1}-\text {a2}},-\frac {\text {c1} \text {a1}^5-\left (\text {a3} \text {b2}^2+\text {a3} \text {b2}+\text {b1} (\text {b2}+1)-\text {c0}+2 \text {a3} \text {c1}+\text {a2} \left (\text {b2}^2+\text {b2}+2 \text {c1}\right )\right ) \text {a1}^4+\left ((\text {b2}+\text {c1}) \text {a2}^2+\left (-\text {b2} \text {b1}+\text {b1}-2 \text {c0}+2 \text {a3} \left (\text {b2}^2+2 \text {b2}+2 \text {c1}\right )\right ) \text {a2}-\text {b1}^2+\text {a3} \text {b1}+\text {a3}^2 \text {b2}-\text {a3} \text {b1} \text {b2}-2 \text {b0} (\text {b2}+1)-2 \text {a3} \text {c0}+\text {a3}^2 \text {c1}\right ) \text {a1}^3-\left ((3 \text {a3} \text {b2}-\text {c0}+2 \text {a3} \text {c1}) \text {a2}^2-3 \text {b0} \text {a2}+\text {a3} (3 \text {a3} \text {b2}-3 \text {b1} \text {b2}-4 \text {c0}+2 \text {a3} \text {c1}) \text {a2}-3 \text {a3} \text {b0}+3 \text {b0} \text {b1}-\text {a3}^2 \text {c0}\right ) \text {a1}^2-\left ((\text {b0}+\text {a3} (\text {b1}+2 \text {c0}-\text {a3} (2 \text {b2}+\text {c1}))) \text {a2}^2+\left ((\text {b1}+2 \text {c0}) \text {a3}^2-\left (\text {b1}^2+2 \text {b0} (\text {b2}-2)\right ) \text {a3}-\text {b0} \text {b1}\right ) \text {a2}+\text {b0} \left (\text {a3}^2-\text {b1} \text {a3}+2 \text {b0}\right )\right ) \text {a1}+\text {a3} \text {b0}^2+\text {a2} \text {b0} \left (\text {a3}^2+\text {b1} \text {a3}+\text {b0}\right )+\text {a2}^2 \text {a3} (\text {b0}+\text {a3} (\text {b1}+\text {c0}))}{(\text {a1}-\text {a2})^3 (\text {a1}-\text {a3})^2},\frac {(\text {b2}+1) \text {a1}^2+(2 \text {b1}+\text {a2} (\text {b2}-1)+\text {a3} (\text {b2}-1)) \text {a1}+\text {a2} \text {a3}+2 \text {b0}-\text {a2} \text {a3} \text {b2}+(\text {a1}-\text {a2}) (\text {a1}-\text {a3}) \sqrt {\text {b2}^2+2 \text {b2}+4 \text {c1}+1}}{2 (\text {a1}-\text {a2}) (\text {a1}-\text {a3})},-\frac {-\left ((\text {a2}-\text {a3}) \left (\text {b2}^2+3 \text {b2}+4 \text {c1}+2\right ) \text {a1}^3\right )+\left (\left (\text {b2}^2+\text {b2}+4 \text {c1}+2\right ) \text {a2}^2-2 \text {b1} \text {a2}+\text {a3} \left (\text {b2}^2+3 \text {b2}+4 \text {c1}+2\right ) \text {a2}-\text {a3} \text {b1} (\text {b2}-1)-\text {b0} (\text {b2}+1)-\text {a3}^2 \left (3 \text {b2}^2+5 \text {b2}+8 \text {c1}+4\right )\right ) \text {a1}^2+\left (2 (\text {b2}+2 \text {c1}+1) \text {a3}^3+(2 \text {a2} (\text {b2}+2 \text {c1}+1)-\text {b1} (3 \text {b2}+1)) \text {a3}^2+\left (-2 \left (\text {b2}^2+\text {b2}+4 \text {c1}+2\right ) \text {a2}^2-\text {b1} (\text {b2}-3) \text {a2}-2 \text {b1}^2+3 \text {b0}-\text {b0} \text {b2}\right ) \text {a3}-\text {b0} (\text {b2} \text {a2}+\text {a2}+2 \text {b1})\right ) \text {a1}-2 \text {b0}^2+\text {a3} \text {b0} (\text {b2} \text {a2}+\text {a2}-2 \text {b1})-2 \text {a2} \text {a3}^3 (\text {b2}+2 \text {c1}+1)+\text {a3}^2 \left (\left (\text {b2}^2+\text {b2}+4 \text {c1}+2\right ) \text {a2}^2+\text {b1} (\text {b2}-1) \text {a2}-2 \text {b0} (\text {b2}+1)\right )+(\text {a1}-\text {a3}) \left ((\text {a2}-\text {a3}) (\text {b2}+2) \text {a1}^2+\left ((\text {b2}-2) \text {a2}^2+2 \text {b1} \text {a2}+2 \text {a3}^2+\text {b0}-\text {a3} \text {b1}\right ) \text {a1}-2 \text {a2} \text {a3}^2+\text {a2} \text {b0}-\text {a3} \left ((\text {b2}-2) \text {a2}^2+\text {b1} \text {a2}+2 \text {b0}\right )\right ) \sqrt {\text {b2}^2+2 \text {b2}+4 \text {c1}+1}}{2 (\text {a1}-\text {a2}) (\text {a1}-\text {a3}) \left ((\text {b2}+1) \text {a3}^2-(\text {a1}+\text {a2}-\text {b1}) \text {a3}+\text {a1} \text {a2}+\text {b0}+(\text {a1}-\text {a3}) (\text {a3}-\text {a2}) \sqrt {\text {b2}^2+2 \text {b2}+4 \text {c1}+1}\right )},\frac {(\text {b2}+2) \text {a1}^2+(-2 \text {a2}-2 \text {a3}+\text {b1}) \text {a1}+2 \text {a2} \text {a3}+\text {b0}}{(\text {a1}-\text {a2}) (\text {a1}-\text {a3})},\frac {\text {b0}+\text {a2} (\text {b1}+\text {a2} \text {b2})}{(\text {a1}-\text {a2}) (\text {a2}-\text {a3})},\frac {\text {a1}-x}{\text {a1}-\text {a2}}\right ] (x-\text {a1})^{\frac {(\text {b2}+1) \text {a1}^2-(\text {a2}+\text {a3}-\text {b1}) \text {a1}+\text {a2} \text {a3}+\text {b0}}{(\text {a1}-\text {a2}) (\text {a1}-\text {a3})}}+c_1 \text {HeunG}\left [\frac {\text {a1}-\text {a3}}{\text {a1}-\text {a2}},-\frac {\text {c0}+\text {a1} \text {c1}}{\text {a1}-\text {a2}},\frac {1}{2} \left (-\text {b2}+\sqrt {\text {b2}^2+2 \text {b2}+4 \text {c1}+1}-1\right ),-\frac {-4 \text {c1} \text {a3}^2+\text {b1} \text {a3}+\text {a2} \left (\text {b2}^2+\text {b2}+4 \text {c1}\right ) \text {a3}+\text {b0}+(\text {b0}+\text {a3} \text {b1}) \text {b2}-\text {a1} (\text {a2}-\text {a3}) \left (\text {b2}^2+\text {b2}+4 \text {c1}\right )+(\text {b0}-\text {a1} \text {a2} \text {b2}+\text {a3} (\text {b1}+(\text {a1}+\text {a2}) \text {b2})) \sqrt {\text {b2}^2+2 \text {b2}+4 \text {c1}+1}}{2 \left (\text {b2} \text {a3}^2+\text {a3}^2-\text {a2} \text {a3}+\text {b1} \text {a3}+\text {a1} (\text {a2}-\text {a3})+\text {b0}+(\text {a1}-\text {a3}) (\text {a3}-\text {a2}) \sqrt {\text {b2}^2+2 \text {b2}+4 \text {c1}+1}\right )},-\frac {\text {b0}+\text {a1} (\text {b1}+\text {a1} \text {b2})}{(\text {a1}-\text {a2}) (\text {a1}-\text {a3})},\frac {\text {b0}+\text {a2} (\text {b1}+\text {a2} \text {b2})}{(\text {a1}-\text {a2}) (\text {a2}-\text {a3})},\frac {\text {a1}-x}{\text {a1}-\text {a2}}\right ]\right \}\right \}\]
Maple ✓
cpu = 2.342 (sec), leaf count = 1123
\[\left [y \left (x \right ) = \textit {\_C1} \mathit {HG}\left (\frac {-\mathit {a3} +\mathit {a1}}{\mathit {a1} -\mathit {a2}}, \frac {-\mathit {a1} \mathit {c1} -\mathit {c0}}{\mathit {a1} -\mathit {a2}}, -\frac {\mathit {b2}}{2}-\frac {1}{2}+\frac {\sqrt {\mathit {b2}^{2}+2 \mathit {b2} +4 \mathit {c1} +1}}{2}, \frac {\left (\left (\mathit {a1} \mathit {b2} +\mathit {b2} \mathit {a2} +\mathit {b1} \right ) \mathit {a3} -\mathit {a1} \mathit {a2} \mathit {b2} +\mathit {b0} \right ) \sqrt {\mathit {b2}^{2}+2 \mathit {b2} +4 \mathit {c1} +1}-4 \mathit {a3}^{2} \mathit {c1} +\left (\left (\mathit {b2}^{2}+\mathit {b2} +4 \mathit {c1} \right ) \mathit {a1} +\left (\mathit {b2}^{2}+\mathit {b2} +4 \mathit {c1} \right ) \mathit {a2} +\mathit {b1} \left (\mathit {b2} +1\right )\right ) \mathit {a3} -\left (\mathit {b2}^{2}+\mathit {b2} +4 \mathit {c1} \right ) \mathit {a2} \mathit {a1} +\mathit {b0} \left (\mathit {b2} +1\right )}{2 \left (-\mathit {a3} +\mathit {a2} \right ) \left (-\mathit {a3} +\mathit {a1} \right ) \sqrt {\mathit {b2}^{2}+2 \mathit {b2} +4 \mathit {c1} +1}+\left (-2 \mathit {b2} -2\right ) \mathit {a3}^{2}+\left (2 \mathit {a1} +2 \mathit {a2} -2 \mathit {b1} \right ) \mathit {a3} -2 \mathit {a1} \mathit {a2} -2 \mathit {b0}}, \frac {-\mathit {b2} \,\mathit {a1}^{2}-\mathit {a1} \mathit {b1} -\mathit {b0}}{\left (\mathit {a1} -\mathit {a2} \right ) \left (-\mathit {a3} +\mathit {a1} \right )}, \frac {\mathit {b2} \,\mathit {a2}^{2}+\mathit {b1} \mathit {a2} +\mathit {b0}}{\left (\mathit {a1} -\mathit {a2} \right ) \left (-\mathit {a3} +\mathit {a2} \right )}, \frac {\mathit {a1} -x}{\mathit {a1} -\mathit {a2}}\right )+\textit {\_C2} \mathit {HG}\left (\frac {-\mathit {a3} +\mathit {a1}}{\mathit {a1} -\mathit {a2}}, \frac {-\left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} \left (\mathit {a1} -2 \mathit {a2} \right ) \mathit {b2} -\mathit {b1} \mathit {a2} -\mathit {b0} +\left (-\mathit {a1} \mathit {c1} -\mathit {c0} \right ) \mathit {a2} +\mathit {a1}^{2} \mathit {c1} +\mathit {c0} \mathit {a1} \right ) \mathit {a3}^{2}+\left (\mathit {a1}^{3} \left (\mathit {a1} -2 \mathit {a2} \right ) \mathit {b2}^{2}+\left (\mathit {a1} \left (\mathit {a1} -3 \mathit {a2} \right ) \mathit {b1} -2 \mathit {a2} \mathit {b0} +\mathit {a1} \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -3 \mathit {a2} \right )\right ) \mathit {a1} \mathit {b2} -\mathit {b1}^{2} \mathit {a2} \mathit {a1} -\left (\mathit {a1} +\mathit {a2} \right ) \left (\mathit {b0} +\mathit {a1} \left (\mathit {a1} -\mathit {a2} \right )\right ) \mathit {b1} -\mathit {b0}^{2}+\left (-3 \mathit {a1}^{2}+4 \mathit {a1} \mathit {a2} -\mathit {a2}^{2}\right ) \mathit {b0} +2 \mathit {a1} \left (\mathit {a1} -\mathit {a2} \right )^{2} \left (\mathit {a1} \mathit {c1} +\mathit {c0} \right )\right ) \mathit {a3} +\mathit {a1}^{4} \mathit {a2} \,\mathit {b2}^{2}+\left (\left (\mathit {a1} +\mathit {a2} \right ) \mathit {b1} +\mathit {a1} \mathit {a2} -\mathit {a2}^{2}+2 \mathit {b0} \right ) \mathit {a1}^{3} \mathit {b2} +\mathit {a1}^{3} \mathit {b1}^{2}+\left (\left (3 \mathit {a1} -\mathit {a2} \right ) \mathit {b0} +\mathit {a1}^{2} \left (\mathit {a1} -\mathit {a2} \right )\right ) \mathit {a1} \mathit {b1} +\left (2 \mathit {a1} -\mathit {a2} \right ) \mathit {b0}^{2}+2 \left (\mathit {a1} -\mathit {a2} \right ) \left (\mathit {a1} -\frac {\mathit {a2}}{2}\right ) \mathit {a1} \mathit {b0} -\mathit {a1}^{2} \left (\mathit {a1} -\mathit {a2} \right )^{2} \left (\mathit {a1} \mathit {c1} +\mathit {c0} \right )}{\left (\mathit {a1} -\mathit {a2} \right )^{3} \left (-\mathit {a3} +\mathit {a1} \right )^{2}}, \frac {\left (-\mathit {a3} +\mathit {a1} \right ) \left (\mathit {a1} -\mathit {a2} \right ) \sqrt {\mathit {b2}^{2}+2 \mathit {b2} +4 \mathit {c1} +1}+\left (\mathit {b2} +1\right ) \mathit {a1}^{2}+\left (\left (\mathit {b2} -1\right ) \mathit {a2} +\mathit {a3} \left (\mathit {b2} -1\right )+2 \mathit {b1} \right ) \mathit {a1} -\mathit {a3} \left (\mathit {b2} -1\right ) \mathit {a2} +2 \mathit {b0}}{2 \left (-\mathit {a3} +\mathit {a1} \right ) \left (\mathit {a1} -\mathit {a2} \right )}, \frac {\left (-\mathit {a3} +\mathit {a1} \right ) \left (\left (\mathit {b2} +2\right ) \left (-\mathit {a3} +\mathit {a2} \right ) \mathit {a1}^{2}+\left (2 \mathit {a3}^{2}-\mathit {b1} \mathit {a3} +\left (\mathit {b2} -2\right ) \mathit {a2}^{2}+2 \mathit {b1} \mathit {a2} +\mathit {b0} \right ) \mathit {a1} -2 \mathit {a2} \,\mathit {a3}^{2}+\left (\left (-\mathit {b2} +2\right ) \mathit {a2}^{2}-\mathit {b1} \mathit {a2} -2 \mathit {b0} \right ) \mathit {a3} +\mathit {a2} \mathit {b0} \right ) \sqrt {\mathit {b2}^{2}+2 \mathit {b2} +4 \mathit {c1} +1}-\left (\mathit {b2}^{2}+3 \mathit {b2} +4 \mathit {c1} +2\right ) \left (-\mathit {a3} +\mathit {a2} \right ) \mathit {a1}^{3}+\left (\left (-3 \mathit {b2}^{2}-5 \mathit {b2} -8 \mathit {c1} -4\right ) \mathit {a3}^{2}+\left (\left (\mathit {b2}^{2}+3 \mathit {b2} +4 \mathit {c1} +2\right ) \mathit {a2} -\left (\mathit {b2} -1\right ) \mathit {b1} \right ) \mathit {a3} +\left (\mathit {b2}^{2}+\mathit {b2} +4 \mathit {c1} +2\right ) \mathit {a2}^{2}-2 \mathit {b1} \mathit {a2} -\mathit {b0} \left (\mathit {b2} +1\right )\right ) \mathit {a1}^{2}+\left (\left (2 \mathit {b2} +4 \mathit {c1} +2\right ) \mathit {a3}^{3}+\left (\left (2 \mathit {b2} +4 \mathit {c1} +2\right ) \mathit {a2} -3 \mathit {b1} \mathit {b2} -\mathit {b1} \right ) \mathit {a3}^{2}+\left (\left (-2 \mathit {b2}^{2}-2 \mathit {b2} -8 \mathit {c1} -4\right ) \mathit {a2}^{2}-\mathit {b1} \left (\mathit {b2} -3\right ) \mathit {a2} -\mathit {b2} \mathit {b0} -2 \mathit {b1}^{2}+3 \mathit {b0} \right ) \mathit {a3} -\mathit {b0} \left (\left (\mathit {b2} +1\right ) \mathit {a2} +2 \mathit {b1} \right )\right ) \mathit {a1} -2 \mathit {a2} \left (\mathit {b2} +2 \mathit {c1} +1\right ) \mathit {a3}^{3}+\left (\left (\mathit {b2}^{2}+\mathit {b2} +4 \mathit {c1} +2\right ) \mathit {a2}^{2}+\left (\mathit {b2} -1\right ) \mathit {b1} \mathit {a2} -2 \mathit {b0} \left (\mathit {b2} +1\right )\right ) \mathit {a3}^{2}+\left (\left (\mathit {b2} +1\right ) \mathit {a2} -2 \mathit {b1} \right ) \mathit {b0} \mathit {a3} -2 \mathit {b0}^{2}}{2 \left (\mathit {a1} -\mathit {a2} \right ) \left (\left (-\mathit {a3} +\mathit {a2} \right ) \left (-\mathit {a3} +\mathit {a1} \right ) \sqrt {\mathit {b2}^{2}+2 \mathit {b2} +4 \mathit {c1} +1}+\left (-\mathit {b2} -1\right ) \mathit {a3}^{2}+\left (\mathit {a1} +\mathit {a2} -\mathit {b1} \right ) \mathit {a3} -\mathit {a1} \mathit {a2} -\mathit {b0} \right ) \left (-\mathit {a3} +\mathit {a1} \right )}, \frac {\left (\mathit {b2} +2\right ) \mathit {a1}^{2}+\left (-2 \mathit {a2} -2 \mathit {a3} +\mathit {b1} \right ) \mathit {a1} +2 \mathit {a2} \mathit {a3} +\mathit {b0}}{\left (-\mathit {a3} +\mathit {a1} \right ) \left (\mathit {a1} -\mathit {a2} \right )}, \frac {\mathit {b2} \,\mathit {a2}^{2}+\mathit {b1} \mathit {a2} +\mathit {b0}}{\left (\mathit {a1} -\mathit {a2} \right ) \left (-\mathit {a3} +\mathit {a2} \right )}, \frac {\mathit {a1} -x}{\mathit {a1} -\mathit {a2}}\right ) \left (-\mathit {a1} +x \right )^{\frac {\left (\mathit {b2} +1\right ) \mathit {a1}^{2}+\left (-\mathit {a2} -\mathit {a3} +\mathit {b1} \right ) \mathit {a1} +\mathit {a2} \mathit {a3} +\mathit {b0}}{\left (-\mathit {a3} +\mathit {a1} \right ) \left (\mathit {a1} -\mathit {a2} \right )}}\right ]\] Mathematica raw input
DSolve[(c0 + c1*x)*y[x] + (b0 + b1*x + b2*x^2)*y'[x] + (a1 - x)*(a2 - x)*(a3 - x)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*HeunG[(a1 - a3)/(a1 - a2), -((c0 + a1*c1)/(a1 - a2)), (-1 - b2 + Sqrt[1 + 2*b2 + b2^2 + 4*c1])/2, -1/2*(b0 + a3*b1 + (b0 + a3*b1)*b2 - 4*a3^2*c1 - a1*(a2 - a3)*(b2 + b2^2 + 4*c1) + a2*a3*(b2 + b2^2 + 4*c1) + (b0 - a1*a2*b2 + a3*(b1 + (a1 + a2)*b2))*Sqrt[1 + 2*b2 + b2^2 + 4*c1])/(a1*(a2 - a3) - a2*a3 + a3^2 + b0 + a3*b1 + a3^2*b2 + (a1 - a3)*(-a2 + a3)*Sqrt[1 + 2*b2 + b2^2 + 4*c1]), -((b0 + a1*(b1 + a1*b2))/((a1 - a2)*(a1 - a3))), (b0 + a2*(b1 + a2*b2))/((a1 - a2)*(a2 - a3)), (a1 - x)/(a1 - a2)] + (-a1 + x)^((a2*a3 + b0 - a1*(a2 + a3 - b1) + a1^2*(1 + b2))/((a1 - a2)*(a1 - a3)))*C[2]*HeunG[(a1 - a3)/(a1 - a2), -((a3*b0^2 + a2*b0*(a3^2 + b0 + a3*b1) + a2^2*a3*(b0 + a3*(b1 + c0)) + a1^5*c1 - a1^4*(a3*b2 + a3*b2^2 + b1*(1 + b2) - c0 + 2*a3*c1 + a2*(b2 + b2^2 + 2*c1)) - a1^2*(-3*a2*b0 - 3*a3*b0 + 3*b0*b1 - a3^2*c0 + a2*a3*(3*a3*b2 - 3*b1*b2 - 4*c0 + 2*a3*c1) + a2^2*(3*a3*b2 - c0 + 2*a3*c1)) + a1^3*(a3*b1 - b1^2 + a3^2*b2 - a3*b1*b2 - 2*b0*(1 + b2) - 2*a3*c0 + a3^2*c1 + a2^2*(b2 + c1) + a2*(b1 - b1*b2 - 2*c0 + 2*a3*(2*b2 + b2^2 + 2*c1))) - a1*(b0*(a3^2 + 2*b0 - a3*b1) + a2*(-(b0*b1) - a3*(b1^2 + 2*b0*(-2 + b2)) + a3^2*(b1 + 2*c0)) + a2^2*(b0 + a3*(b1 + 2*c0 - a3*(2*b2 + c1)))))/((a1 - a2)^3*(a1 - a3)^2)), (a2*a3 + 2*b0 + a1*(2*b1 + a2*(-1 + b2) + a3*(-1 + b2)) - a2*a3*b2 + a1^2*(1 + b2) + (a1 - a2)*(a1 - a3)*Sqrt[1 + 2*b2 + b2^2 + 4*c1])/(2*(a1 - a2)*(a1 - a3)), -1/2*(-2*b0^2 + a3*b0*(a2 - 2*b1 + a2*b2) - 2*a2*a3^3*(1 + b2 + 2*c1) + (a1 - a3)*(-2*a2*a3^2 + a2*b0 - a3*(2*b0 + a2*b1 + a2^2*(-2 + b2)) + a1*(2*a3^2 + b0 + 2*a2*b1 - a3*b1 + a2^2*(-2 + b2)) + a1^2*(a2 - a3)*(2 + b2))*Sqrt[1 + 2*b2 + b2^2 + 4*c1] - a1^3*(a2 - a3)*(2 + 3*b2 + b2^2 + 4*c1) + a3^2*(a2*b1*(-1 + b2) - 2*b0*(1 + b2) + a2^2*(2 + b2 + b2^2 + 4*c1)) + a1^2*(-2*a2*b1 - a3*b1*(-1 + b2) - b0*(1 + b2) + a2^2*(2 + b2 + b2^2 + 4*c1) + a2*a3*(2 + 3*b2 + b2^2 + 4*c1) - a3^2*(4 + 5*b2 + 3*b2^2 + 8*c1)) + a1*(-(b0*(a2 + 2*b1 + a2*b2)) + 2*a3^3*(1 + b2 + 2*c1) + a3^2*(-(b1*(1 + 3*b2)) + 2*a2*(1 + b2 + 2*c1)) + a3*(3*b0 - 2*b1^2 - a2*b1*(-3 + b2) - b0*b2 - 2*a2^2*(2 + b2 + b2^2 + 4*c1))))/((a1 - a2)*(a1 - a3)*(a1*a2 + b0 - a3*(a1 + a2 - b1) + a3^2*(1 + b2) + (a1 - a3)*(-a2 + a3)*Sqrt[1 + 2*b2 + b2^2 + 4*c1])), (2*a2*a3 + b0 + a1*(-2*a2 - 2*a3 + b1) + a1^2*(2 + b2))/((a1 - a2)*(a1 - a3)), (b0 + a2*(b1 + a2*b2))/((a1 - a2)*(a2 - a3)), (a1 - x)/(a1 - a2)]}}
Maple raw input
dsolve((a1-x)*(a2-x)*(a3-x)*diff(diff(y(x),x),x)+(b2*x^2+b1*x+b0)*diff(y(x),x)+(c1*x+c0)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*HeunG((-a3+a1)/(a1-a2),(-a1*c1-c0)/(a1-a2),-1/2*b2-1/2+1/2*(b2^2+2*b2+4*c1+1)^(1/2),(((a1*b2+a2*b2+b1)*a3-a1*a2*b2+b0)*(b2^2+2*b2+4*c1+1)^(1/2)-4*a3^2*c1+((b2^2+b2+4*c1)*a1+(b2^2+b2+4*c1)*a2+b1*(b2+1))*a3-(b2^2+b2+4*c1)*a2*a1+b0*(b2+1))/(2*(-a3+a2)*(-a3+a1)*(b2^2+2*b2+4*c1+1)^(1/2)+(-2*b2-2)*a3^2+(2*a1+2*a2-2*b1)*a3-2*a1*a2-2*b0),(-a1^2*b2-a1*b1-b0)/(a1-a2)/(-a3+a1),(a2^2*b2+a2*b1+b0)/(a1-a2)/(-a3+a2),(a1-x)/(a1-a2))+_C2*HeunG((-a3+a1)/(a1-a2),(-(a1-a2)*(a1*(a1-2*a2)*b2-b1*a2-b0+(-a1*c1-c0)*a2+a1^2*c1+c0*a1)*a3^2+(a1^3*(a1-2*a2)*b2^2+(a1*(a1-3*a2)*b1-2*a2*b0+a1*(a1-a2)*(a1-3*a2))*a1*b2-b1^2*a2*a1-(a1+a2)*(b0+a1*(a1-a2))*b1-b0^2+(-3*a1^2+4*a1*a2-a2^2)*b0+2*a1*(a1-a2)^2*(a1*c1+c0))*a3+a1^4*a2*b2^2+((a1+a2)*b1+a1*a2-a2^2+2*b0)*a1^3*b2+a1^3*b1^2+((3*a1-a2)*b0+a1^2*(a1-a2))*a1*b1+(2*a1-a2)*b0^2+2*(a1-a2)*(a1-1/2*a2)*a1*b0-a1^2*(a1-a2)^2*(a1*c1+c0))/(a1-a2)^3/(-a3+a1)^2,1/2*((-a3+a1)*(a1-a2)*(b2^2+2*b2+4*c1+1)^(1/2)+(b2+1)*a1^2+((b2-1)*a2+a3*(b2-1)+2*b1)*a1-a3*(b2-1)*a2+2*b0)/(-a3+a1)/(a1-a2),1/2*((-a3+a1)*((b2+2)*(-a3+a2)*a1^2+(2*a3^2-b1*a3+(b2-2)*a2^2+2*b1*a2+b0)*a1-2*a2*a3^2+((-b2+2)*a2^2-b1*a2-2*b0)*a3+a2*b0)*(b2^2+2*b2+4*c1+1)^(1/2)-(b2^2+3*b2+4*c1+2)*(-a3+a2)*a1^3+((-3*b2^2-5*b2-8*c1-4)*a3^2+((b2^2+3*b2+4*c1+2)*a2-(b2-1)*b1)*a3+(b2^2+b2+4*c1+2)*a2^2-2*b1*a2-b0*(b2+1))*a1^2+((2*b2+4*c1+2)*a3^3+((2*b2+4*c1+2)*a2-3*b1*b2-b1)*a3^2+((-2*b2^2-2*b2-8*c1-4)*a2^2-b1*(b2-3)*a2-b2*b0-2*b1^2+3*b0)*a3-b0*((b2+1)*a2+2*b1))*a1-2*a2*(b2+2*c1+1)*a3^3+((b2^2+b2+4*c1+2)*a2^2+(b2-1)*b1*a2-2*b0*(b2+1))*a3^2+((b2+1)*a2-2*b1)*b0*a3-2*b0^2)/(a1-a2)/((-a3+a2)*(-a3+a1)*(b2^2+2*b2+4*c1+1)^(1/2)+(-b2-1)*a3^2+(a1+a2-b1)*a3-a1*a2-b0)/(-a3+a1),((b2+2)*a1^2+(-2*a2-2*a3+b1)*a1+2*a2*a3+b0)/(-a3+a1)/(a1-a2),(a2^2*b2+a2*b1+b0)/(a1-a2)/(-a3+a2),(a1-x)/(a1-a2))*(-a1+x)^(((b2+1)*a1^2+(-a2-a3+b1)*a1+a2*a3+b0)/(-a3+a1)/(a1-a2))]