Internal problem ID [8635]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1055.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\mathit {a1} \,x^{2}+\mathit {b1} x +\mathit {c1} \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 317
dsolve(diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)+(a1*x^2+b1*x+c1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \hypergeom \left (\left [\frac {\left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}+a^{3}-2 a^{2} \mathit {c1} +\left (2 \mathit {b1} b -4 \mathit {a1} \right ) a +\left (-2 b^{2}+8 \mathit {c1} \right ) \mathit {a1} -2 \mathit {b1}^{2}}{4 \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {\left (a^{2} x +b a -4 \mathit {a1} x -2 \mathit {b1} \right )^{2}}{2 \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {x \left (\left (a x +2 b \right ) \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}+\left (a^{2}-4 \mathit {a1} \right ) \left (a^{2} x +2 b a -4 \mathit {a1} x -4 \mathit {b1} \right )\right )}{4 \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}}}+c_{2} \left (a^{2} x +b a -4 \mathit {a1} x -2 \mathit {b1} \right ) \hypergeom \left (\left [\frac {3 \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}+a^{3}-2 a^{2} \mathit {c1} +\left (2 \mathit {b1} b -4 \mathit {a1} \right ) a +\left (-2 b^{2}+8 \mathit {c1} \right ) \mathit {a1} -2 \mathit {b1}^{2}}{4 \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (a^{2} x +b a -4 \mathit {a1} x -2 \mathit {b1} \right )^{2}}{2 \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {x \left (\left (a x +2 b \right ) \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}+\left (a^{2}-4 \mathit {a1} \right ) \left (a^{2} x +2 b a -4 \mathit {a1} x -4 \mathit {b1} \right )\right )}{4 \left (a^{2}-4 \mathit {a1} \right )^{\frac {3}{2}}}} \]
✓ Solution by Mathematica
Time used: 0.082 (sec). Leaf size: 304
DSolve[(c1 + b1*x + a1*x^2)*y[x] + (b + a*x)*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \exp \left (-\frac {x \left (2 b \sqrt {a^2-4 \text {a1}}+a x \sqrt {a^2-4 \text {a1}}+a^2 x+2 a b-4 (\text {a1} x+\text {b1})\right )}{4 \sqrt {a^2-4 \text {a1}}}\right ) \left (c_1 \text {HermiteH}\left (\frac {-a^3+2 \left (\text {a1} \left (2 \sqrt {a^2-4 \text {a1}}+b^2-4 \text {c1}\right )+\text {b1}^2\right )-a^2 \left (\sqrt {a^2-4 \text {a1}}-2 \text {c1}\right )+a (4 \text {a1}-2 b \text {b1})}{2 \left (a^2-4 \text {a1}\right )^{3/2}},\frac {a^2 x+a b-2 (2 \text {a1} x+\text {b1})}{\sqrt {2} \left (a^2-4 \text {a1}\right )^{3/4}}\right )+c_2 \, _1F_1\left (\frac {a^3+\left (\sqrt {a^2-4 \text {a1}}-2 \text {c1}\right ) a^2+(2 b \text {b1}-4 \text {a1}) a-2 \left (\text {b1}^2+\text {a1} \left (b^2-4 \text {c1}+2 \sqrt {a^2-4 \text {a1}}\right )\right )}{4 \left (a^2-4 \text {a1}\right )^{3/2}};\frac {1}{2};\frac {\left (x a^2+b a-2 (\text {b1}+2 \text {a1} x)\right )^2}{2 \left (a^2-4 \text {a1}\right )^{3/2}}\right )\right ) \\ \end{align*}