ODE
\[ (\text {a0}+\text {b0} x) y'(x)+y(x) \left (\text {a1}+\text {b1} x+\text {c1} x^2\right )+y''(x)=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.287362 (sec), leaf count = 303
\[\left \{\left \{y(x)\to \exp \left (-\frac {x \left (2 \text {a0} \left (\sqrt {\text {b0}^2-4 \text {c1}}+\text {b0}\right )+x \left (\text {b0} \sqrt {\text {b0}^2-4 \text {c1}}+\text {b0}^2-4 \text {c1}\right )-4 \text {b1}\right )}{4 \sqrt {\text {b0}^2-4 \text {c1}}}\right ) \left (c_1 H_{\frac {-\text {b0}^3-\sqrt {\text {b0}^2-4 \text {c1}} \text {b0}^2-2 \text {a0} \text {b1} \text {b0}+4 \text {c1} \text {b0}+2 \text {b1}^2+2 \text {a1} \left (\text {b0}^2-4 \text {c1}\right )+2 \text {a0}^2 \text {c1}+4 \sqrt {\text {b0}^2-4 \text {c1}} \text {c1}}{2 \left (\text {b0}^2-4 \text {c1}\right )^{3/2}}}\left (\frac {x \text {b0}^2+\text {a0} \text {b0}-2 \text {b1}-4 \text {c1} x}{\sqrt {2} \left (\text {b0}^2-4 \text {c1}\right )^{3/4}}\right )+c_2 \, _1F_1\left (\frac {\text {b0}^3-2 \text {a1} \text {b0}^2+\sqrt {\text {b0}^2-4 \text {c1}} \text {b0}^2+2 \text {a0} \text {b1} \text {b0}-4 \text {c1} \text {b0}-2 \text {b1}^2-2 \text {a0}^2 \text {c1}+8 \text {a1} \text {c1}-4 \sqrt {\text {b0}^2-4 \text {c1}} \text {c1}}{4 \left (\text {b0}^2-4 \text {c1}\right )^{3/2}};\frac {1}{2};\frac {\left (x \text {b0}^2+\text {a0} \text {b0}-2 \text {b1}-4 \text {c1} x\right )^2}{2 \left (\text {b0}^2-4 \text {c1}\right )^{3/2}}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.683 (sec), leaf count = 313
\[\left [y \left (x \right ) = \textit {\_C1} \hypergeom \left (\left [-\frac {-\frac {\left (\mathit {b0}^{2}-4 \mathit {c1} \right )^{\frac {3}{2}}}{2}-\frac {\mathit {b0}^{3}}{2}+\mathit {a1} \,\mathit {b0}^{2}+\left (-\mathit {a0} \mathit {b1} +2 \mathit {c1} \right ) \mathit {b0} +\left (\mathit {a0}^{2}-4 \mathit {a1} \right ) \mathit {c1} +\mathit {b1}^{2}}{2 \left (\mathit {b0}^{2}-4 \mathit {c1} \right )^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {\left (x \,\mathit {b0}^{2}+\mathit {a0} \mathit {b0} -4 \mathit {c1} x -2 \mathit {b1} \right )^{2}}{2 \left (\mathit {b0}^{2}-4 \mathit {c1} \right )^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {\left (\left (\mathit {b0} x +2 \mathit {a0} \right ) \left (\mathit {b0}^{2}-4 \mathit {c1} \right )^{\frac {3}{2}}+\left (\mathit {b0}^{2}-4 \mathit {c1} \right ) \left (x \,\mathit {b0}^{2}+2 \mathit {a0} \mathit {b0} -4 \mathit {c1} x -4 \mathit {b1} \right )\right ) x}{4 \left (\mathit {b0}^{2}-4 \mathit {c1} \right )^{\frac {3}{2}}}}+\textit {\_C2} \left (x \,\mathit {b0}^{2}+\mathit {a0} \mathit {b0} -4 \mathit {c1} x -2 \mathit {b1} \right ) \hypergeom \left (\left [-\frac {-\frac {3 \left (\mathit {b0}^{2}-4 \mathit {c1} \right )^{\frac {3}{2}}}{2}-\frac {\mathit {b0}^{3}}{2}+\mathit {a1} \,\mathit {b0}^{2}+\left (-\mathit {a0} \mathit {b1} +2 \mathit {c1} \right ) \mathit {b0} +\left (\mathit {a0}^{2}-4 \mathit {a1} \right ) \mathit {c1} +\mathit {b1}^{2}}{2 \left (\mathit {b0}^{2}-4 \mathit {c1} \right )^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (x \,\mathit {b0}^{2}+\mathit {a0} \mathit {b0} -4 \mathit {c1} x -2 \mathit {b1} \right )^{2}}{2 \left (\mathit {b0}^{2}-4 \mathit {c1} \right )^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {\left (\left (\mathit {b0} x +2 \mathit {a0} \right ) \left (\mathit {b0}^{2}-4 \mathit {c1} \right )^{\frac {3}{2}}+\left (\mathit {b0}^{2}-4 \mathit {c1} \right ) \left (x \,\mathit {b0}^{2}+2 \mathit {a0} \mathit {b0} -4 \mathit {c1} x -4 \mathit {b1} \right )\right ) x}{4 \left (\mathit {b0}^{2}-4 \mathit {c1} \right )^{\frac {3}{2}}}}\right ]\] Mathematica raw input
DSolve[(a1 + b1*x + c1*x^2)*y[x] + (a0 + b0*x)*y'[x] + y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (C[1]*HermiteH[(-b0^3 - 2*a0*b0*b1 + 2*b1^2 - b0^2*Sqrt[b0^2 - 4*c1] + 2*a1*(b0^2 - 4*c1) + 2*a0^2*c1 + 4*b0*c1 + 4*Sqrt[b0^2 - 4*c1]*c1)/(2*(b0^2 - 4*c1)^(3/2)), (a0*b0 - 2*b1 + b0^2*x - 4*c1*x)/(Sqrt[2]*(b0^2 - 4*c1)^(3/4))] + C[2]*Hypergeometric1F1[(-2*a1*b0^2 + b0^3 + 2*a0*b0*b1 - 2*b1^2 + b0^2*Sqrt[b0^2 - 4*c1] - 2*a0^2*c1 + 8*a1*c1 - 4*b0*c1 - 4*Sqrt[b0^2 - 4*c1]*c1)/(4*(b0^2 - 4*c1)^(3/2)), 1/2, (a0*b0 - 2*b1 + b0^2*x - 4*c1*x)^2/(2*(b0^2 - 4*c1)^(3/2))])/E^((x*(-4*b1 + 2*a0*(b0 + Sqrt[b0^2 - 4*c1]) + (b0^2 + b0*Sqrt[b0^2 - 4*c1] - 4*c1)*x))/(4*Sqrt[b0^2 - 4*c1]))}}
Maple raw input
dsolve(diff(diff(y(x),x),x)+(b0*x+a0)*diff(y(x),x)+(c1*x^2+b1*x+a1)*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*hypergeom([-1/2/(b0^2-4*c1)^(3/2)*(-1/2*(b0^2-4*c1)^(3/2)-1/2*b0^3+a1*b0^2+(-a0*b1+2*c1)*b0+(a0^2-4*a1)*c1+b1^2)],[1/2],1/2*(b0^2*x+a0*b0-4*c1*x-2*b1)^2/(b0^2-4*c1)^(3/2))*exp(-1/4*((b0*x+2*a0)*(b0^2-4*c1)^(3/2)+(b0^2-4*c1)*(b0^2*x+2*a0*b0-4*c1*x-4*b1))/(b0^2-4*c1)^(3/2)*x)+_C2*(b0^2*x+a0*b0-4*c1*x-2*b1)*hypergeom([-1/2*(-3/2*(b0^2-4*c1)^(3/2)-1/2*b0^3+a1*b0^2+(-a0*b1+2*c1)*b0+(a0^2-4*a1)*c1+b1^2)/(b0^2-4*c1)^(3/2)],[3/2],1/2*(b0^2*x+a0*b0-4*c1*x-2*b1)^2/(b0^2-4*c1)^(3/2))*exp(-1/4*((b0*x+2*a0)*(b0^2-4*c1)^(3/2)+(b0^2-4*c1)*(b0^2*x+2*a0*b0-4*c1*x-4*b1))/(b0^2-4*c1)^(3/2)*x)]