ODE
\[ x (\text {a1}+\text {b1} x) y'(x)+y(x) (\text {a2}+\text {b2} x)+(1-x) x^2 y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.523465 (sec), leaf count = 317
\[\left \{\left \{y(x)\to i^{-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}-\text {a1}+1} x^{\frac {1}{2} \left (-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}-\text {a1}+1\right )} \left (c_1 \, _2F_1\left (\frac {1}{2} \left (-\text {a1}-\text {b1}-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}-\sqrt {\text {b1}^2+2 \text {b1}+4 \text {b2}+1}\right ),\frac {1}{2} \left (-\text {a1}-\text {b1}-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+\sqrt {\text {b1}^2+2 \text {b1}+4 \text {b2}+1}\right );1-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1};x\right )+c_2 i^{2 \sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}} x^{\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}} \, _2F_1\left (\frac {1}{2} \left (-\text {a1}-\text {b1}+\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}-\sqrt {\text {b1}^2+2 \text {b1}+4 \text {b2}+1}\right ),\frac {1}{2} \left (-\text {a1}-\text {b1}+\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+\sqrt {\text {b1}^2+2 \text {b1}+4 \text {b2}+1}\right );\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+1;x\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.159 (sec), leaf count = 241
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{-{\frac {{\it a1}}{2}}+{\frac {1}{2}\sqrt {{{\it a1}}^{2}-2\,{\it a1}-4\,{\it a2}+1}}+{\frac {1}{2}}}{\mbox {$_2$F$_1$}(-{\frac {{\it b1}}{2}}-{\frac {{\it a1}}{2}}+{\frac {1}{2}\sqrt {{{\it a1}}^{2}-2\,{\it a1}-4\,{\it a2}+1}}+{\frac {1}{2}\sqrt {{{\it b1}}^{2}+2\,{\it b1}+4\,{\it b2}+1}},-{\frac {{\it b1}}{2}}-{\frac {{\it a1}}{2}}+{\frac {1}{2}\sqrt {{{\it a1}}^{2}-2\,{\it a1}-4\,{\it a2}+1}}-{\frac {1}{2}\sqrt {{{\it b1}}^{2}+2\,{\it b1}+4\,{\it b2}+1}};\,1+\sqrt {{{\it a1}}^{2}-2\,{\it a1}-4\,{\it a2}+1};\,x)}+{\it \_C2}\,{x}^{-{\frac {{\it a1}}{2}}-{\frac {1}{2}\sqrt {{{\it a1}}^{2}-2\,{\it a1}-4\,{\it a2}+1}}+{\frac {1}{2}}}{\mbox {$_2$F$_1$}(-{\frac {{\it b1}}{2}}-{\frac {{\it a1}}{2}}-{\frac {1}{2}\sqrt {{{\it a1}}^{2}-2\,{\it a1}-4\,{\it a2}+1}}+{\frac {1}{2}\sqrt {{{\it b1}}^{2}+2\,{\it b1}+4\,{\it b2}+1}},-{\frac {{\it b1}}{2}}-{\frac {{\it a1}}{2}}-{\frac {1}{2}\sqrt {{{\it a1}}^{2}-2\,{\it a1}-4\,{\it a2}+1}}-{\frac {1}{2}\sqrt {{{\it b1}}^{2}+2\,{\it b1}+4\,{\it b2}+1}};\,1-\sqrt {{{\it a1}}^{2}-2\,{\it a1}-4\,{\it a2}+1};\,x)} \right \} \] Mathematica raw input
DSolve[(a2 + b2*x)*y[x] + x*(a1 + b1*x)*y'[x] + (1 - x)*x^2*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> I^(1 - a1 - Sqrt[1 - 2*a1 + a1^2 - 4*a2])*x^((1 - a1 - Sqrt[1 - 2*a1 + a1^2 - 4*a2])/2)*(C[1]*Hypergeometric2F1[(-a1 - Sqrt[1 - 2*a1 + a1^2 - 4*a2] - b1 - Sqrt[1 + 2*b1 + b1^2 + 4*b2])/2, (-a1 - Sqrt[1 - 2*a1 + a1^2 - 4*a2] - b1 + Sqrt[1 + 2*b1 + b1^2 + 4*b2])/2, 1 - Sqrt[1 - 2*a1 + a1^2 - 4*a2], x] + I^(2*Sqrt[1 - 2*a1 + a1^2 - 4*a2])*x^Sqrt[1 - 2*a1 + a1^2 - 4*a2]*C[2]*Hypergeometric2F1[(-a1 + Sqrt[1 - 2*a1 + a1^2 - 4*a2] - b1 - Sqrt[1 + 2*b1 + b1^2 + 4*b2])/2, (-a1 + Sqrt[1 - 2*a1 + a1^2 - 4*a2] - b1 + Sqrt[1 + 2*b1 + b1^2 + 4*b2])/2, 1 + Sqrt[1 - 2*a1 + a1^2 - 4*a2], x])}}
Maple raw input
dsolve(x^2*(1-x)*diff(diff(y(x),x),x)+x*(b1*x+a1)*diff(y(x),x)+(b2*x+a2)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*x^(-1/2*a1+1/2*(a1^2-2*a1-4*a2+1)^(1/2)+1/2)*hypergeom([-1/2*b1-1/2*a1+1/2*(a1^2-2*a1-4*a2+1)^(1/2)+1/2*(b1^2+2*b1+4*b2+1)^(1/2), -1/2*b1-1/2*a1+1/2*(a1^2-2*a1-4*a2+1)^(1/2)-1/2*(b1^2+2*b1+4*b2+1)^(1/2)],[1+(a1^2-2*a1-4*a2+1)^(1/2)],x)+_C2*x^(-1/2*a1-1/2*(a1^2-2*a1-4*a2+1)^(1/2)+1/2)*hypergeom([-1/2*b1-1/2*a1-1/2*(a1^2-2*a1-4*a2+1)^(1/2)+1/2*(b1^2+2*b1+4*b2+1)^(1/2), -1/2*b1-1/2*a1-1/2*(a1^2-2*a1-4*a2+1)^(1/2)-1/2*(b1^2+2*b1+4*b2+1)^(1/2)],[1-(a1^2-2*a1-4*a2+1)^(1/2)],x)