ODE
\[ x (\text {a0}+x) y''(x)+(\text {a1}+\text {b1} x) y'(x)+\text {a2} y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.266966 (sec), leaf count = 160
\[\left \{\left \{y(x)\to c_2 \text {a0}^{\frac {\text {a1}}{\text {a0}}-1} x^{1-\frac {\text {a1}}{\text {a0}}} \, _2F_1\left (\frac {\text {a0} \left (\text {b1}-\sqrt {(\text {b1}-1)^2-4 \text {a2}}+1\right )-2 \text {a1}}{2 \text {a0}},\frac {\text {a0} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {a2}}+1\right )-2 \text {a1}}{2 \text {a0}};2-\frac {\text {a1}}{\text {a0}};-\frac {x}{\text {a0}}\right )+c_1 \, _2F_1\left (\frac {1}{2} \left (\text {b1}-\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right ),\frac {1}{2} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right );\frac {\text {a1}}{\text {a0}};-\frac {x}{\text {a0}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.15 (sec), leaf count = 249
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}(-{\frac {1}{2}}+{\frac {{\it b1}}{2}}-{\frac {1}{2}\sqrt {{{\it b1}}^{2}-4\,{\it a2}-2\,{\it b1}+1}},-{\frac {1}{2}}+{\frac {{\it b1}}{2}}+{\frac {1}{2}\sqrt {{{\it b1}}^{2}-4\,{\it a2}-2\,{\it b1}+1}};\,{\frac {1}{2} \left ( {\it b1}\,\sqrt {{{\it a0}}^{2}}+{\it b1}\,{\it a0}-2\,{\it a1} \right ) {\frac {1}{\sqrt {{{\it a0}}^{2}}}}};\,{\frac {1}{2} \left ( \sqrt {{{\it a0}}^{2}}+{\it a0}+2\,x \right ) {\frac {1}{\sqrt {{{\it a0}}^{2}}}}})}+{\it \_C2}\, \left ( \sqrt {{{\it a0}}^{2}}+{\it a0}+2\,x \right ) ^{-{\frac {1}{2\,{{\it a0}}^{2}} \left ( \left ( {\it b1}-2 \right ) \sqrt {{{\it a0}}^{2}}+{\it b1}\,{\it a0}-2\,{\it a1} \right ) \sqrt {{{\it a0}}^{2}}}}{\mbox {$_2$F$_1$}(-{\frac {1}{2} \left ( \sqrt {{{\it b1}}^{2}-4\,{\it a2}-2\,{\it b1}+1}\sqrt {{{\it a0}}^{2}}-\sqrt {{{\it a0}}^{2}}+{\it b1}\,{\it a0}-2\,{\it a1} \right ) {\frac {1}{\sqrt {{{\it a0}}^{2}}}}},{\frac {1}{2} \left ( \sqrt {{{\it a0}}^{2}}+\sqrt {{{\it b1}}^{2}-4\,{\it a2}-2\,{\it b1}+1}\sqrt {{{\it a0}}^{2}}-{\it b1}\,{\it a0}+2\,{\it a1} \right ) {\frac {1}{\sqrt {{{\it a0}}^{2}}}}};\,-{\frac {1}{2} \left ( \left ( {\it b1}-4 \right ) \sqrt {{{\it a0}}^{2}}+{\it b1}\,{\it a0}-2\,{\it a1} \right ) {\frac {1}{\sqrt {{{\it a0}}^{2}}}}};\,{\frac {1}{2} \left ( \sqrt {{{\it a0}}^{2}}+{\it a0}+2\,x \right ) {\frac {1}{\sqrt {{{\it a0}}^{2}}}}})} \right \} \] Mathematica raw input
DSolve[a2*y[x] + (a1 + b1*x)*y'[x] + x*(a0 + x)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1]*Hypergeometric2F1[(-1 - Sqrt[-4*a2 + (-1 + b1)^2] + b1)/2, (-1 + Sqrt[-4*a2 + (-1 + b1)^2] + b1)/2, a1/a0, -(x/a0)] + a0^(-1 + a1/a0)*x^(1 - a1/a0)*C[2]*Hypergeometric2F1[(-2*a1 + a0*(1 - Sqrt[-4*a2 + (-1 + b1)^2] + b1))/(2*a0), (-2*a1 + a0*(1 + Sqrt[-4*a2 + (-1 + b1)^2] + b1))/(2*a0), 2 - a1/a0, -(x/a0)]}}
Maple raw input
dsolve(x*(a0+x)*diff(diff(y(x),x),x)+(b1*x+a1)*diff(y(x),x)+a2*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*hypergeom([-1/2+1/2*b1-1/2*(b1^2-4*a2-2*b1+1)^(1/2), -1/2+1/2*b1+1/2*(b1^2-4*a2-2*b1+1)^(1/2)],[1/2*(b1*(a0^2)^(1/2)+b1*a0-2*a1)/(a0^2)^(1/2)],1/2/(a0^2)^(1/2)*((a0^2)^(1/2)+a0+2*x))+_C2*((a0^2)^(1/2)+a0+2*x)^(-1/2*((b1-2)*(a0^2)^(1/2)+b1*a0-2*a1)*(a0^2)^(1/2)/a0^2)*hypergeom([-1/2/(a0^2)^(1/2)*((b1^2-4*a2-2*b1+1)^(1/2)*(a0^2)^(1/2)-(a0^2)^(1/2)+b1*a0-2*a1), 1/2/(a0^2)^(1/2)*((a0^2)^(1/2)+(b1^2-4*a2-2*b1+1)^(1/2)*(a0^2)^(1/2)-b1*a0+2*a1)],[-1/2/(a0^2)^(1/2)*((b1-4)*(a0^2)^(1/2)+b1*a0-2*a1)],1/2/(a0^2)^(1/2)*((a0^2)^(1/2)+a0+2*x))