ODE
\[ y''(x) \left (\text {a0}+\text {b0} x+\text {c0} x^2\right )+(\text {a1}+\text {b1} x) y'(x)+2 \text {a2} y(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 4.88339 (sec), leaf count = 498
\[\left \{\left \{y(x)\to c_1 \, _2F_1\left (\frac {\text {b1}-\text {c0}+\sqrt {(\text {b1}-\text {c0})^2-8 \text {a2} \text {c0}}}{2 \text {c0}},-\frac {-\text {b1}+\text {c0}+\sqrt {(\text {b1}-\text {c0})^2-8 \text {a2} \text {c0}}}{2 \text {c0}};\frac {\text {b1} \left (\text {b0}+\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}\right )-2 \text {a1} \text {c0}}{2 \text {c0} \sqrt {\text {b0}^2-4 \text {a0} \text {c0}}};\frac {\text {b0}+2 \text {c0} x+\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}{2 \sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}\right )-c_2 2^{-\frac {\text {a1}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}+\frac {\frac {\text {b0} \text {b1}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}+\text {b1}}{2 \text {c0}}-1} \exp \left (-\frac {i \pi \left (\text {b1} \left (\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}+\text {b0}\right )-2 \text {a1} \text {c0}\right )}{2 \text {c0} \sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}\right ) \left (\frac {\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}+\text {b0}+2 \text {c0} x}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}\right )^{\frac {\text {a1}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}-\frac {\frac {\text {b0} \text {b1}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}+\text {b1}}{2 \text {c0}}+1} \, _2F_1\left (\frac {-\frac {\text {b0} \text {b1}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}+\text {c0}-\sqrt {(\text {b1}-\text {c0})^2-8 \text {a2} \text {c0}}+\frac {2 \text {a1} \text {c0}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}}{2 \text {c0}},\frac {-\frac {\text {b0} \text {b1}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}+\text {c0}+\sqrt {(\text {b1}-\text {c0})^2-8 \text {a2} \text {c0}}+\frac {2 \text {a1} \text {c0}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}}{2 \text {c0}};-\frac {\frac {\text {b0} \text {b1}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}+\text {b1}-4 \text {c0}-\frac {2 \text {a1} \text {c0}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}}{2 \text {c0}};\frac {\text {b0}+2 \text {c0} x+\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}{2 \sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}\right )\right \}\right \}\]
Maple ✓
cpu = 0.844 (sec), leaf count = 507
\[\left [y \left (x \right ) = \textit {\_C1} \hypergeom \left (\left [-\frac {\mathit {c0} -\mathit {b1} +\sqrt {\mathit {c0}^{2}+\left (-8 \mathit {a2} -2 \mathit {b1} \right ) \mathit {c0} +\mathit {b1}^{2}}}{2 \mathit {c0}}, \frac {-\mathit {c0} +\mathit {b1} +\sqrt {\mathit {c0}^{2}+\left (-8 \mathit {a2} -2 \mathit {b1} \right ) \mathit {c0} +\mathit {b1}^{2}}}{2 \mathit {c0}}\right ], \left [\frac {\mathit {b1} \sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}\, \mathit {c0} -2 \mathit {c0} \mathit {a1} +\mathit {b1} \mathit {b0}}{2 \mathit {c0}^{2} \sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}}\right ], \frac {-2 \sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}\, x \,\mathit {c0}^{2}-\sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}\, \mathit {b0} \mathit {c0} +4 \mathit {a0} \mathit {c0} -\mathit {b0}^{2}}{8 \mathit {a0} \mathit {c0} -2 \mathit {b0}^{2}}\right )+\textit {\_C2} \left (2 \sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}\, x \,\mathit {c0}^{2}+\sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}\, \mathit {b0} \mathit {c0} -4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}\right )^{\frac {-\frac {\mathit {c0} \left (\mathit {b1} -2 \mathit {c0} \right ) \sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}}{2}+\mathit {c0} \mathit {a1} -\frac {\mathit {b1} \mathit {b0}}{2}}{\sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}\, \mathit {c0}^{2}}} \hypergeom \left (\left [\frac {\frac {\mathit {c0} \left (\mathit {c0} +\sqrt {\mathit {c0}^{2}+\left (-8 \mathit {a2} -2 \mathit {b1} \right ) \mathit {c0} +\mathit {b1}^{2}}\right ) \sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}}{2}+\mathit {c0} \mathit {a1} -\frac {\mathit {b1} \mathit {b0}}{2}}{\sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}\, \mathit {c0}^{2}}, \frac {\frac {\mathit {c0} \left (\mathit {c0} -\sqrt {\mathit {c0}^{2}+\left (-8 \mathit {a2} -2 \mathit {b1} \right ) \mathit {c0} +\mathit {b1}^{2}}\right ) \sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}}{2}+\mathit {c0} \mathit {a1} -\frac {\mathit {b1} \mathit {b0}}{2}}{\sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}\, \mathit {c0}^{2}}\right ], \left [\frac {-\frac {\mathit {c0} \left (\mathit {b1} -4 \mathit {c0} \right ) \sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}}{2}+\mathit {c0} \mathit {a1} -\frac {\mathit {b1} \mathit {b0}}{2}}{\sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}\, \mathit {c0}^{2}}\right ], \frac {-2 \sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}\, x \,\mathit {c0}^{2}-\sqrt {\frac {-4 \mathit {a0} \mathit {c0} +\mathit {b0}^{2}}{\mathit {c0}^{2}}}\, \mathit {b0} \mathit {c0} +4 \mathit {a0} \mathit {c0} -\mathit {b0}^{2}}{8 \mathit {a0} \mathit {c0} -2 \mathit {b0}^{2}}\right )\right ]\] Mathematica raw input
DSolve[2*a2*y[x] + (a1 + b1*x)*y'[x] + (a0 + b0*x + c0*x^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((2^(-1 - a1/Sqrt[b0^2 - 4*a0*c0] + (b1 + (b0*b1)/Sqrt[b0^2 - 4*a0*c0])/(2*c0))*((b0 + Sqrt[b0^2 - 4*a0*c0] + 2*c0*x)/Sqrt[b0^2 - 4*a0*c0])^(1 + a1/Sqrt[b0^2 - 4*a0*c0] - (b1 + (b0*b1)/Sqrt[b0^2 - 4*a0*c0])/(2*c0))*C[2]*Hypergeometric2F1[(c0 - (b0*b1)/Sqrt[b0^2 - 4*a0*c0] + (2*a1*c0)/Sqrt[b0^2 - 4*a0*c0] - Sqrt[(b1 - c0)^2 - 8*a2*c0])/(2*c0), (c0 - (b0*b1)/Sqrt[b0^2 - 4*a0*c0] + (2*a1*c0)/Sqrt[b0^2 - 4*a0*c0] + Sqrt[(b1 - c0)^2 - 8*a2*c0])/(2*c0), -1/2*(b1 - 4*c0 + (b0*b1)/Sqrt[b0^2 - 4*a0*c0] - (2*a1*c0)/Sqrt[b0^2 - 4*a0*c0])/c0, (b0 + Sqrt[b0^2 - 4*a0*c0] + 2*c0*x)/(2*Sqrt[b0^2 - 4*a0*c0])])/E^(((I/2)*(-2*a1*c0 + b1*(b0 + Sqrt[b0^2 - 4*a0*c0]))*Pi)/(c0*Sqrt[b0^2 - 4*a0*c0]))) + C[1]*Hypergeometric2F1[(b1 - c0 + Sqrt[(b1 - c0)^2 - 8*a2*c0])/(2*c0), -1/2*(-b1 + c0 + Sqrt[(b1 - c0)^2 - 8*a2*c0])/c0, (-2*a1*c0 + b1*(b0 + Sqrt[b0^2 - 4*a0*c0]))/(2*c0*Sqrt[b0^2 - 4*a0*c0]), (b0 + Sqrt[b0^2 - 4*a0*c0] + 2*c0*x)/(2*Sqrt[b0^2 - 4*a0*c0])]}}
Maple raw input
dsolve((c0*x^2+b0*x+a0)*diff(diff(y(x),x),x)+(b1*x+a1)*diff(y(x),x)+2*a2*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*hypergeom([-1/2/c0*(c0-b1+(c0^2+(-8*a2-2*b1)*c0+b1^2)^(1/2)), 1/2/c0*(-c0+b1+(c0^2+(-8*a2-2*b1)*c0+b1^2)^(1/2))],[1/2*(b1*((-4*a0*c0+b0^2)/c0^2)^(1/2)*c0-2*c0*a1+b1*b0)/c0^2/((-4*a0*c0+b0^2)/c0^2)^(1/2)],(-2*((-4*a0*c0+b0^2)/c0^2)^(1/2)*x*c0^2-((-4*a0*c0+b0^2)/c0^2)^(1/2)*b0*c0+4*a0*c0-b0^2)/(8*a0*c0-2*b0^2))+_C2*(2*((-4*a0*c0+b0^2)/c0^2)^(1/2)*x*c0^2+((-4*a0*c0+b0^2)/c0^2)^(1/2)*b0*c0-4*a0*c0+b0^2)^((-1/2*c0*(b1-2*c0)*((-4*a0*c0+b0^2)/c0^2)^(1/2)+c0*a1-1/2*b1*b0)/((-4*a0*c0+b0^2)/c0^2)^(1/2)/c0^2)*hypergeom([1/((-4*a0*c0+b0^2)/c0^2)^(1/2)*(1/2*c0*(c0+(c0^2+(-8*a2-2*b1)*c0+b1^2)^(1/2))*((-4*a0*c0+b0^2)/c0^2)^(1/2)+c0*a1-1/2*b1*b0)/c0^2, (1/2*c0*(c0-(c0^2+(-8*a2-2*b1)*c0+b1^2)^(1/2))*((-4*a0*c0+b0^2)/c0^2)^(1/2)+c0*a1-1/2*b1*b0)/((-4*a0*c0+b0^2)/c0^2)^(1/2)/c0^2],[(-1/2*c0*(b1-4*c0)*((-4*a0*c0+b0^2)/c0^2)^(1/2)+c0*a1-1/2*b1*b0)/((-4*a0*c0+b0^2)/c0^2)^(1/2)/c0^2],(-2*((-4*a0*c0+b0^2)/c0^2)^(1/2)*x*c0^2-((-4*a0*c0+b0^2)/c0^2)^(1/2)*b0*c0+4*a0*c0-b0^2)/(8*a0*c0-2*b0^2))]