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 = 0.316035 (sec), leaf count = 156
\[\left \{\left \{y(x)\to c_2 \text {a0}^{\frac {\text {a1}}{\text {a0}}-1} x^{1-\frac {\text {a1}}{\text {a0}}} \, _2F_1\left (\frac {1}{2} \left (\text {b1}-\sqrt {(\text {b1}-1)^2-4 \text {a2}}+1\right )-\frac {\text {a1}}{\text {a0}},\frac {1}{2} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {a2}}+1\right )-\frac {\text {a1}}{\text {a0}};2-\frac {\text {a1}}{\text {a0}};-\frac {x}{\text {a0}}\right )+c_1 \, _2F_1\left (\frac {1}{2} \left (\text {b1}-\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right ),\frac {1}{2} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {a2}}-1\right );\frac {\text {a1}}{\text {a0}};-\frac {x}{\text {a0}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.585 (sec), leaf count = 249
\[\left [y \left (x \right ) = \textit {\_C1} \hypergeom \left (\left [-\frac {1}{2}+\frac {\mathit {b1}}{2}-\frac {\sqrt {\mathit {b1}^{2}-4 \mathit {a2} -2 \mathit {b1} +1}}{2}, -\frac {1}{2}+\frac {\mathit {b1}}{2}+\frac {\sqrt {\mathit {b1}^{2}-4 \mathit {a2} -2 \mathit {b1} +1}}{2}\right ], \left [\frac {\mathit {b1} \sqrt {\mathit {a0}^{2}}+\mathit {a0} \mathit {b1} -2 \mathit {a1}}{2 \sqrt {\mathit {a0}^{2}}}\right ], \frac {\sqrt {\mathit {a0}^{2}}+\mathit {a0} +2 x}{2 \sqrt {\mathit {a0}^{2}}}\right )+\textit {\_C2} \left (\sqrt {\mathit {a0}^{2}}+\mathit {a0} +2 x \right )^{-\frac {\left (\left (\mathit {b1} -2\right ) \sqrt {\mathit {a0}^{2}}+\mathit {a0} \mathit {b1} -2 \mathit {a1} \right ) \sqrt {\mathit {a0}^{2}}}{2 \mathit {a0}^{2}}} \hypergeom \left (\left [-\frac {\sqrt {\mathit {b1}^{2}-4 \mathit {a2} -2 \mathit {b1} +1}\, \sqrt {\mathit {a0}^{2}}-\sqrt {\mathit {a0}^{2}}+\mathit {a0} \mathit {b1} -2 \mathit {a1}}{2 \sqrt {\mathit {a0}^{2}}}, \frac {\sqrt {\mathit {a0}^{2}}+\sqrt {\mathit {b1}^{2}-4 \mathit {a2} -2 \mathit {b1} +1}\, \sqrt {\mathit {a0}^{2}}-\mathit {a0} \mathit {b1} +2 \mathit {a1}}{2 \sqrt {\mathit {a0}^{2}}}\right ], \left [-\frac {\left (\mathit {b1} -4\right ) \sqrt {\mathit {a0}^{2}}+\mathit {a0} \mathit {b1} -2 \mathit {a1}}{2 \sqrt {\mathit {a0}^{2}}}\right ], \frac {\sqrt {\mathit {a0}^{2}}+\mathit {a0} +2 x}{2 \sqrt {\mathit {a0}^{2}}}\right )\right ]\] 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[-(a1/a0) + (1 - Sqrt[-4*a2 + (-1 + b1)^2] + b1)/2, -(a1/a0) + (1 + Sqrt[-4*a2 + (-1 + b1)^2] + b1)/2, 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))
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)+a0*b1-2*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)+a0*b1-2*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)+a0*b1-2*a1), 1/2/(a0^2)^(1/2)*((a0^2)^(1/2)+(b1^2-4*a2-2*b1+1)^(1/2)*(a0^2)^(1/2)-a0*b1+2*a1)],[-1/2/(a0^2)^(1/2)*((b1-4)*(a0^2)^(1/2)+a0*b1-2*a1)],1/2/(a0^2)^(1/2)*((a0^2)^(1/2)+a0+2*x))]