ODE
\[ x^2 (\text {a0}+x) y''(x)+x (\text {a1}+\text {b1} x) y'(x)+y(x) (\text {a2}+\text {b2} x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.643332 (sec), leaf count = 391
\[\left \{\left \{y(x)\to \text {a0}^{-\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}+\text {a0}-\text {a1}}{2 \text {a0}}} x^{-\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}-\text {a0}+\text {a1}}{2 \text {a0}}} \left (c_2 x^{\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}}{\text {a0}}} \, _2F_1\left (\frac {-\text {a1}+\text {a0} \left (\text {b1}+\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}\right )+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{2 \text {a0}},\frac {\text {b1} \text {a0}-\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1} \text {a0}-\text {a1}+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{2 \text {a0}};\frac {\text {a0}+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{\text {a0}};-\frac {x}{\text {a0}}\right )+c_1 \text {a0}^{\frac {\sqrt {\text {a0}^2-2 \text {a0} (\text {a1}+2 \text {a2})+\text {a1}^2}}{\text {a0}}} \, _2F_1\left (-\frac {\text {a1}-\text {a0} \left (\text {b1}+\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1}\right )+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{2 \text {a0}},-\frac {-\text {b1} \text {a0}+\sqrt {\text {b1}^2-2 \text {b1}-4 \text {b2}+1} \text {a0}+\text {a1}+\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{2 \text {a0}};1-\frac {\sqrt {\text {a0}^2-2 (\text {a1}+2 \text {a2}) \text {a0}+\text {a1}^2}}{\text {a0}};-\frac {x}{\text {a0}}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.731 (sec), leaf count = 361
\[\left [y \left (x \right ) = \textit {\_C1} \,x^{\frac {-\mathit {a1} +\sqrt {\mathit {a0}^{2}+\left (-2 \mathit {a1} -4 \mathit {a2} \right ) \mathit {a0} +\mathit {a1}^{2}}+\mathit {a0}}{2 \mathit {a0}}} \hypergeom \left (\left [\frac {-\mathit {a0} \mathit {b1} -\sqrt {\mathit {b1}^{2}-2 \mathit {b1} -4 \mathit {b2} +1}\, \mathit {a0} +2 \mathit {a0} +\mathit {a1} +\sqrt {\mathit {a0}^{2}+\left (-2 \mathit {a1} -4 \mathit {a2} \right ) \mathit {a0} +\mathit {a1}^{2}}}{2 \mathit {a0}}, \frac {-\mathit {a0} \mathit {b1} +\sqrt {\mathit {b1}^{2}-2 \mathit {b1} -4 \mathit {b2} +1}\, \mathit {a0} +2 \mathit {a0} +\mathit {a1} +\sqrt {\mathit {a0}^{2}+\left (-2 \mathit {a1} -4 \mathit {a2} \right ) \mathit {a0} +\mathit {a1}^{2}}}{2 \mathit {a0}}\right ], \left [\frac {\mathit {a0} +\sqrt {\mathit {a0}^{2}+\left (-2 \mathit {a1} -4 \mathit {a2} \right ) \mathit {a0} +\mathit {a1}^{2}}}{\mathit {a0}}\right ], -\frac {x}{\mathit {a0}}\right ) \left (\mathit {a0} +x \right )^{\frac {-\mathit {a0} \mathit {b1} +\mathit {a0} +\mathit {a1}}{\mathit {a0}}}+\textit {\_C2} \,x^{\frac {-\mathit {a1} -\sqrt {\mathit {a0}^{2}+\left (-2 \mathit {a1} -4 \mathit {a2} \right ) \mathit {a0} +\mathit {a1}^{2}}+\mathit {a0}}{2 \mathit {a0}}} \hypergeom \left (\left [\frac {-\mathit {a0} \mathit {b1} +\sqrt {\mathit {b1}^{2}-2 \mathit {b1} -4 \mathit {b2} +1}\, \mathit {a0} +2 \mathit {a0} +\mathit {a1} -\sqrt {\mathit {a0}^{2}+\left (-2 \mathit {a1} -4 \mathit {a2} \right ) \mathit {a0} +\mathit {a1}^{2}}}{2 \mathit {a0}}, -\frac {\sqrt {\mathit {b1}^{2}-2 \mathit {b1} -4 \mathit {b2} +1}\, \mathit {a0} +\mathit {a0} \mathit {b1} +\sqrt {\mathit {a0}^{2}+\left (-2 \mathit {a1} -4 \mathit {a2} \right ) \mathit {a0} +\mathit {a1}^{2}}-2 \mathit {a0} -\mathit {a1}}{2 \mathit {a0}}\right ], \left [\frac {\mathit {a0} -\sqrt {\mathit {a0}^{2}+\left (-2 \mathit {a1} -4 \mathit {a2} \right ) \mathit {a0} +\mathit {a1}^{2}}}{\mathit {a0}}\right ], -\frac {x}{\mathit {a0}}\right ) \left (\mathit {a0} +x \right )^{\frac {-\mathit {a0} \mathit {b1} +\mathit {a0} +\mathit {a1}}{\mathit {a0}}}\right ]\] Mathematica raw input
DSolve[(a2 + b2*x)*y[x] + x*(a1 + b1*x)*y'[x] + x^2*(a0 + x)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (a0^(Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]/a0)*C[1]*Hypergeometric2F1[-1/2*(a1 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)] - a0*(b1 + Sqrt[1 - 2*b1 + b1^2 - 4*b2]))/a0, -1/2*(a1 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)] - a0*b1 + a0*Sqrt[1 - 2*b1 + b1^2 - 4*b2])/a0, 1 - Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]/a0, -(x/a0)] + x^(Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)]/a0)*C[2]*Hypergeometric2F1[(-a1 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)] + a0*(b1 + Sqrt[1 - 2*b1 + b1^2 - 4*b2]))/(2*a0), (-a1 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)] + a0*b1 - a0*Sqrt[1 - 2*b1 + b1^2 - 4*b2])/(2*a0), (a0 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)])/a0, -(x/a0)])/(a0^((a0 - a1 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)])/(2*a0))*x^((-a0 + a1 + Sqrt[a0^2 + a1^2 - 2*a0*(a1 + 2*a2)])/(2*a0)))}}
Maple raw input
dsolve(x^2*(a0+x)*diff(diff(y(x),x),x)+x*(b1*x+a1)*diff(y(x),x)+(b2*x+a2)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*x^(1/2*(-a1+(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2)+a0)/a0)*hypergeom([1/2*(-a0*b1-(b1^2-2*b1-4*b2+1)^(1/2)*a0+2*a0+a1+(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2))/a0, 1/2/a0*(-a0*b1+(b1^2-2*b1-4*b2+1)^(1/2)*a0+2*a0+a1+(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2))],[1/a0*(a0+(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2))],-1/a0*x)*(a0+x)^((-a0*b1+a0+a1)/a0)+_C2*x^(1/2*(-a1-(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2)+a0)/a0)*hypergeom([1/2/a0*(-a0*b1+(b1^2-2*b1-4*b2+1)^(1/2)*a0+2*a0+a1-(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2)), -1/2/a0*((b1^2-2*b1-4*b2+1)^(1/2)*a0+a0*b1+(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2)-2*a0-a1)],[(a0-(a0^2+(-2*a1-4*a2)*a0+a1^2)^(1/2))/a0],-1/a0*x)*(a0+x)^((-a0*b1+a0+a1)/a0)]